# kb

id: list-of-publications-in-mathematics-20260505-6
file: data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-6.md
category: reference
title: List of publications in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_mathematics
date_saved: 2026-05-05T08:16:32.440767+00:00
instance: kb-cron
parent: list-of-publications-in-mathematics-20260505-0
summary:
=== L'anneau d'homologie d'une représentation, Structure de l'anneau d'homologie d'une représentation === Jean Leray (1946) These two Comptes Rendus notes of Leray from 1946 introduced the novel concepts of sheafs, sheaf cohomology, and spectral sequences, which he had developed during his years of captivity as a prisoner of war. Leray's announcements and applications (published in other Comptes Rendus notes from 1946) drew immediate attention from other mathematicians. Subsequent clarification, development, and generalization by Henri Cartan, Jean-Louis Koszul, Armand Borel, Jean-Pierre Serre, and Leray himself allowed these concepts to be understood and applied to many other areas of mathematics. Dieudonné would later write that these notions created by Leray "undoubtedly rank at the same level in the history of mathematics as the methods invented by Poincaré and Brouwer". === Quelques propriétés globales des variétés differentiables === René Thom (1954) In this paper, Thom proved the Thom transversality theorem, introduced the notions of oriented and unoriented cobordism, and demonstrated that cobordism groups could be computed as the homotopy groups of certain Thom spaces. Thom completely characterized the unoriented cobordism ring and achieved strong results for several problems, including Steenrod's problem on the realization of cycles. == Category theory == === "General Theory


id: list-of-publications-in-mathematics-20260505-7
file: data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-7.md
category: reference
title: List of publications in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_mathematics
date_saved: 2026-05-05T08:16:32.440767+00:00
instance: kb-cron
parent: list-of-publications-in-mathematics-20260505-0
summary:
Kurt Gödel (1931) Online version: Online version In mathematical logic, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. The first incompleteness theorem states: For any formal system such that (1) it is ω {\displaystyle \omega } -consistent (omega-consistent), (2) it has a recursively definable set of axioms and rules of derivation, and (3) every recursive relation of natural numbers is definable in it, there exists a formula of the system such that, according to the intended interpretation of the system, it expresses a truth about natural numbers and yet it is not a theorem of the system. === Systems of Logic Based on Ordinals === Alan Turing's PhD thesis (1938) == Combinatorics == === "On sets of integers containing no k elements in arithmetic progression" === Endre Szemerédi (1975) Settled a conjecture of Paul Erdős and Pál Turán (now known as Szemerédi's theorem) that if a sequence of natural numbers has positive upper density then it contains arbitrarily long arithmetic progressions. Szemerédi's solution has been described as a "masterpiece of combinatorics" and it introduced new ideas and tools to the field including a weak form of the Szemerédi regularity lemma. === Graph theory === ====


id: list-of-publications-in-mathematics-20260505-8
file: data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-8.md
category: reference
title: List of publications in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_mathematics
date_saved: 2026-05-05T08:16:32.440767+00:00
instance: kb-cron
parent: list-of-publications-in-mathematics-20260505-0
summary:
==== "Полиномиальный алгоритм в линейном программировании" ==== Khachiyan, Leonid Genrikhovich (1979). Полиномиальный алгоритм в линейном программировании [A polynomial algorithm for linear programming]. Doklady Akademii Nauk SSSR (in Russian). 244: 1093–1096.. Khachiyan's work on the ellipsoid method. This was the first polynomial time algorithm for linear programming. == Early manuscripts == These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the history of mathematics. === Moscow Mathematical Papyrus === This is one of the earliest mathematical treatises that still survives today. The Papyrus contains 25 problems involving arithmetic, geometry, and algebra, each with a solution given. Written in Ancient Egypt at approximately 1850 BC. === Rhind Mathematical Papyrus === Ahmes (scribe) One of the oldest mathematical texts, dating to the Second Intermediate Period of ancient Egypt. It was copied by the scribe Ahmes (properly Ahmose) from an older Middle Kingdom papyrus. It laid the foundations of Egyptian mathematics and in turn, later influenced Greek and Hellenistic mathematics. Besides describing how to obtain an approximation of π only missing the mark by less than one per cent, Ahmes make use of a kind of an analogue of the cotangent. === Archimedes Palimpsest ===


id: list-of-publications-in-mathematics-20260505-9
file: data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-9.md
category: reference
title: List of publications in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_mathematics
date_saved: 2026-05-05T08:16:32.440767+00:00
instance: kb-cron
parent: list-of-publications-in-mathematics-20260505-0
summary:
=== Naive Set Theory === Paul Halmos An undergraduate introduction to not-very-naive set theory which has lasted for decades. It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo–Fraenkel set theory and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like large cardinals. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before. === Cardinal and Ordinal Numbers === Wacław Sierpiński The nec plus ultra reference for basic facts about cardinal and ordinal numbers. If you have a question about the cardinality of sets occurring in everyday mathematics, the first place to look is this book, first published in the early 1950s but based on the author's lectures on the subject over the preceding 40 years. === Set Theory: An Introduction to Independence Proofs === Kenneth Kunen This book is not really


id: list-of-q-analogs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_q-analogs-0.md
category: reference
title: List of q-analogs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_q-analogs
date_saved: 2026-05-05T08:16:33.618494+00:00
instance: kb-cron
parent: list-of-q-analogs-20260505-0
summary:
This is a list of q-analogs in mathematics and related fields. == Algebra == Iwahori–Hecke algebra Quantum affine algebra Quantum enveloping algebra Quantum group == Analysis == Jackson integral q-derivative q-difference polynomial Quantum calculus == Combinatorics == LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity == Orthogonal polynomials == q-Bessel polynomials q-Charlier polynomials q-Hahn polynomials q-Jacobi polynomials: Big q-Jacobi polynomials Continuous q-Jacobi polynomials Little q-Jacobi polynomials q-Krawtchouk polynomials q-Laguerre polynomials Continuous q-Legendre polynomials q-Meixner polynomials q-Meixner–Pollaczek polynomials q-Racah polynomials == Probability and statistics == Gaussian q-distribution q-exponential distribution q-Weibull distribution Tsallis q-Gaussian Tsallis entropy == Special functions == Basic hypergeometric series Elliptic gamma function Hahn–Exton q-Bessel function Jackson q-Bessel function q-exponential q-gamma function q-theta function == See also == Lists of mathematics topics


id: list-of-mathematical-topics-in-quantum-theory-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory-0.md
category: reference
title: List of mathematical topics in quantum theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory
date_saved: 2026-05-05T08:16:34.798624+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-quantum-theory-20260505-0
summary:
This is a list of mathematical topics in quantum theory, by Wikipedia page. See also list of functional analysis topics, list of Lie group topics, list of quantum-mechanical systems with analytical solutions. == Mathematical formulation of quantum mechanics == bra–ket notation canonical commutation relation complete set of commuting observables Heisenberg picture Hilbert space Interaction picture Measurement in quantum mechanics quantum field theory quantum logic quantum operation Schrödinger picture semiclassical statistical ensemble wavefunction wave–particle duality Wightman axioms WKB approximation == Schrödinger equation == quantum mechanics, matrix mechanics, Hamiltonian (quantum mechanics) particle in a box particle in a ring particle in a spherically symmetric potential quantum harmonic oscillator hydrogen atom ring wave guide particle in a one-dimensional lattice (periodic potential) Fock symmetry in theory of hydrogen == Symmetry == identical particles angular momentum angular momentum operator rotational invariance rotational symmetry rotation operator translational symmetry Lorentz symmetry Parity transformation Noether's theorem Noether charge Spin (physics) isospin Aman matrices scale invariance spontaneous symmetry breaking supersymmetry breaking == Quantum states == quantum number Pauli exclusion principle quantum indeterminacy uncertainty principle wavefunction collapse zero-point energy bound state coherent state squeezed coherent state density state Fock state, Fock space vacuum state quasinormal mode no-cloning theorem quantum entanglement


id: list-of-random-number-generators-20260505-0
file: data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md
category: reference
title: List of random number generators
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_random_number_generators
date_saved: 2026-05-05T08:16:35.976218+00:00
instance: kb-cron
parent: list-of-random-number-generators-20260505-0
summary:
Random number generators are important in many kinds of technical applications, including physics, engineering or mathematical computer studies (e.g., Monte Carlo simulations), cryptography and gambling (on game servers). This list includes many common types, regardless of quality or applicability to a given use case. == Pseudorandom number generators (PRNGs) == The following algorithms are pseudorandom number generators. == Cryptographic algorithms == Cipher algorithms and cryptographic hashes can be used as very high-quality pseudorandom number generators. However, generally they are considerably slower (typically by a factor 2–10) than fast, non-cryptographic random number generators. These include: Stream ciphers. Popular choices are Salsa20 or ChaCha (often with the number of rounds reduced to 8 for speed), ISAAC, HC-128 and RC4. Block ciphers in counter mode. Common choices are AES (which is very fast on systems supporting it in hardware), TwoFish, Serpent and Camellia. Cryptographic hash functions A few cryptographically secure pseudorandom number generators do not rely on cipher algorithms but try to link mathematically the difficulty of distinguishing their output from a `true' random stream to a computationally difficult problem. These approaches are theoretically important but are too slow to be practical in most applications. They include: Blum–Micali algorithm (1984) Blum Blum Shub


id: list-of-real-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_real_analysis_topics-0.md
category: reference
title: List of real analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_real_analysis_topics
date_saved: 2026-05-05T08:16:37.250027+00:00
instance: kb-cron
parent: list-of-real-analysis-topics-20260505-0
summary:
This is a list of articles that are considered real analysis topics. See also: glossary of real and complex analysis. == General topics == === Limits === Limit of a sequence Subsequential limit – the limit of some subsequence Limit of a function (see List of limits for a list of limits of common functions) One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below Squeeze theorem – confirms the limit of a function via comparison with two other functions Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions === Sequences and series === (see also list of mathematical series) Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number Harmonic progression – a


id: list-of-real-analysis-topics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_real_analysis_topics-1.md
category: reference
title: List of real analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_real_analysis_topics
date_saved: 2026-05-05T08:16:37.250027+00:00
instance: kb-cron
parent: list-of-real-analysis-topics-20260505-0
summary:
== Fundamental theorems == Monotone convergence theorem – relates monotonicity with convergence Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc Taylor's theorem – gives an approximation of a k {\displaystyle k} times differentiable function around a given point by a k {\displaystyle k} -th order Taylor-polynomial. L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms Abel's theorem – relates the limit of a power series to the sum of its coefficients Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function Darboux's theorem – states that all functions that result from the


id: list-of-properties-of-sets-of-reals-20260505-0
file: data/en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals-0.md
category: reference
title: List of properties of sets of reals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals
date_saved: 2026-05-05T08:16:38.430648+00:00
instance: kb-cron
parent: list-of-properties-of-sets-of-reals-20260505-0
summary:
This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology. == Definability properties == Borel set Analytic set C-measurable set Projective set Inductive set Infinity-Borel set Suslin set Homogeneously Suslin set Weakly homogeneously Suslin set Set of uniqueness == Regularity properties == Property of Baire Lebesgue measurable Universally measurable set Perfect set property Universally Baire set == Largeness and smallness properties == Meager set Comeager set - A comeager set is one whose complement is meager. Null set Conull set Dense set Nowhere dense set


id: list-of-recreational-number-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_recreational_number_theory_topics-0.md
category: reference
title: List of recreational number theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_recreational_number_theory_topics
date_saved: 2026-05-05T08:16:39.664179+00:00
instance: kb-cron
parent: list-of-recreational-number-theory-topics-20260505-0
summary:
This is a list of recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins in challenging problems posed purely for their own sake. See list of number theory topics for pages dealing with aspects of number theory with more consolidated theories. == Number sequences == Integer sequence Fibonacci sequence Golden mean base Fibonacci coding Lucas sequence Padovan sequence Figurate numbers Polygonal number Triangular number Square number Pentagonal number Hexagonal number Heptagonal number Octagonal number Nonagonal number Decagonal number Centered polygonal number Centered square number Centered pentagonal number Centered hexagonal number Tetrahedral number Pyramidal number Triangular pyramidal number Square pyramidal number Pentagonal pyramidal number Hexagonal pyramidal number Heptagonal pyramidal number Octahedral number Star number Perfect number Quasiperfect number Almost perfect number Multiply perfect number Hyperperfect number Semiperfect number Primitive semiperfect number Unitary perfect number Weird number Untouchable number Amicable number Sociable number Abundant number Deficient number Amenable number Aliquot sequence Super-Poulet number Lucky number Powerful number Primeval number Palindromic number Telephone number Triangular square number Harmonic divisor number Sphenic number Smith number Double Mersenne number Zeisel number Heteromecic number Niven numbers Superparticular number Highly composite number Highly totient


id: list-of-mathematics-reference-tables-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_reference_tables-0.md
category: reference
title: List of mathematics reference tables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_reference_tables
date_saved: 2026-05-05T08:16:41.067246+00:00
instance: kb-cron
parent: list-of-mathematics-reference-tables-20260505-0
summary:
See also: List of reference tables == Mathematics == List of mathematical topics List of statistical topics List of mathematical functions List of mathematical theorems List of mathematical proofs List of matrices List of numbers List of relativistic equations List of small groups Mathematical constants Sporadic group Table of Clebsch-Gordan coefficients Table of derivatives Table of divisors Table of integrals Table of mathematical symbols Table of prime factors Taylor series Timeline of mathematics Trigonometric identities Truth table


id: list-of-mathematical-topics-in-relativity-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity-0.md
category: reference
title: List of mathematical topics in relativity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity
date_saved: 2026-05-05T08:16:42.212375+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-relativity-20260505-0
summary:
This is a list of mathematical topics in relativity, by Wikipedia page. == Special relativity == Foundational issues principle of relativity speed of light faster-than-light biquaternion conjugate diameters four-vector four-acceleration four-force four-gradient four-momentum four-velocity hyperbolic orthogonality hyperboloid model light-like Lorentz covariance Lorentz group Lorentz transformation Lorentz–FitzGerald contraction hypothesis Minkowski diagram Minkowski space Poincaré group proper length proper time rapidity relativistic wave equations relativistic mass split-complex number unit hyperbola world line == General relativity == black holes no-hair theorem Hawking radiation Hawking temperature Black hole entropy charged black hole rotating black hole micro black hole Schwarzschild black hole Schwarzschild metric Schwarzschild radius Reissner–Nordström black hole Immirzi parameter closed timelike curve cosmic censorship hypothesis chronology protection conjecture Einstein–Cartan theory Einstein's field equation geodesic gravitational redshift Penrose–Hawking singularity theorems Pseudo-Riemannian manifold stress–energy tensor worm hole == Cosmology == anti-de Sitter space Ashtekar variables Batalin–Vilkovisky formalism Big Bang Cauchy horizon cosmic inflation cosmic microwave background cosmic variance cosmological constant dark energy dark matter de Sitter space Friedmann–Lemaître–Robertson–Walker metric horizon problem large-scale structure of the cosmos Randall–Sundrum model warped geometry Weyl curvature hypothesis


id: list-of-representation-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_representation_theory_topics-0.md
category: reference
title: List of representation theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_representation_theory_topics
date_saved: 2026-05-05T08:16:43.462653+00:00
instance: kb-cron
parent: list-of-representation-theory-topics-20260505-0
summary:
This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. See also: Glossary of representation theory == General representation theory == Linear representation Unitary representation Trivial representation Irreducible representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation Schur's lemma Restricted representation == Representation theory of groups == Group representation Group ring Maschke's theorem Regular representation Character (mathematics) Character theory Class function Representation theory of finite groups Modular representation theory Frobenius reciprocity Restricted representation Induced representation Peter–Weyl theorem Young tableau Spherical harmonic Hecke operator Representation theory of the symmetric group Representation theory of diffeomorphism groups Permutation representation Affine representation Projective representation Central extension == Representation theory of Lie groups and Lie algebras == Representation of a Lie group Lie algebra representation, Representation of a Lie superalgebra Universal enveloping algebra Casimir element Infinitesimal character Harish-Chandra homomorphism Fundamental representation Antifundamental representation Bifundamental representation Adjoint representation Weight (representation theory) Cartan's theorem Spinor Wigner's classification, Representation theory of the Poincaré group Wigner–Eckart theorem Stone–von Neumann theorem Orbit method Kirillov character formula Weyl character formula Discrete series representation Principal series representation Borel–Weil–Bott theorem


id: list-of-formulas-in-riemannian-geometry-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-0.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise. == Christoffel symbols, covariant derivative == In a smooth coordinate chart, the Christoffel symbols of the first kind are given by Γ k i j = 1 2 ( ∂ ∂ x j g k i + ∂ ∂ x i g k j − ∂ ∂ x k g i j ) = 1 2 ( g k i , j + g k j , i − g i j , k ) , {\displaystyle \Gamma _{kij}={\frac {1}{2}}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,} and the Christoffel symbols of the second kind by Γ m i j = g m k Γ k i j = 1 2 g m k ( ∂ ∂ x j g k i + ∂ ∂ x i g k j − ∂ ∂ x k g i j ) = 1 2 g m k ( g k i , j + g k j , i − g i j , k ) . {\displaystyle {\begin{aligned}\Gamma


id: list-of-formulas-in-riemannian-geometry-20260505-1
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-1.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
R i j k l = ∂ Γ j k l ∂ x i − ∂ Γ i k l ∂ x j + ( Γ j k p Γ i p l − Γ i k p Γ j p l ) {\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}-{\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}+{\big (}\Gamma _{jk}^{p}\Gamma _{ip}^{l}-\Gamma _{ik}^{p}\Gamma _{jp}^{l}{\big )}} R ( u , v ) w = ∇ u ∇ v w − ∇ v ∇ u w − ∇ [ u , v ] w {\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w} ==== (3,1) Riemann curvature tensor ==== R j k l i = ∂ Γ l j i ∂ x k − ∂ Γ k j i ∂ x l + ( Γ k p i Γ l j p − Γ l p i Γ k j p ) {\displaystyle {R_{jkl}^{i}}={\frac {\partial \Gamma _{lj}^{i}}{\partial x^{k}}}-{\frac {\partial \Gamma _{kj}^{i}}{\partial x^{l}}}+{\big (}\Gamma _{kp}^{i}\Gamma _{lj}^{p}-\Gamma _{lp}^{i}\Gamma _{kj}^{p}{\big )}} ==== Ricci curvature ==== R i k = R i j k j {\displaystyle R_{ik}={R_{ijk}}^{j}} Ric ⁡ ( v , w ) = tr ⁡ ( u ↦ R ( u , v ) w ) {\displaystyle \operatorname {Ric} (v,w)=\operatorname {tr} (u\mapsto


id: list-of-formulas-in-riemannian-geometry-20260505-2
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-2.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
div g ⁡ Ric = 1 2 d R {\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR} Equivalently: g p q ∇ p G q k = 0 {\displaystyle g^{pq}\nabla _{p}G_{qk}=0} div g ⁡ G = 0 {\displaystyle \operatorname {div} _{g}G=0} ==== Ricci identity ==== If X {\displaystyle X} is a vector field then ∇ i ∇ j X k − ∇ j ∇ i X k = − R i j p k X p , {\displaystyle \nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},} which is just the definition of the Riemann tensor. If ω {\displaystyle \omega } is a one-form then ∇ i ∇ j ω k − ∇ j ∇ i ω k = R i j k p ω p . {\displaystyle \nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.} More generally, if T {\displaystyle T} is a (0,k)-tensor field then ∇ i ∇ j T l 1 ⋯ l k − ∇ j ∇ i T l 1 ⋯ l k = R i j l 1 p T p l 2 ⋯ l k + ⋯ + R i j l k p T l 1 ⋯ l k − 1 p . {\displaystyle \nabla _{i}\nabla _{j}T_{l_{1}\cdots


id: list-of-formulas-in-riemannian-geometry-20260505-3
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-3.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
R i k ℓ m = 1 2 ( ∂ 2 g i m ∂ x k ∂ x ℓ + ∂ 2 g k ℓ ∂ x i ∂ x m − ∂ 2 g i ℓ ∂ x k ∂ x m − ∂ 2 g k m ∂ x i ∂ x ℓ ) {\displaystyle R_{ik\ell m}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{\ell }}}+{\frac {\partial ^{2}g_{k\ell }}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{i\ell }}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{\ell }}}\right)} R ℓ i j k = ∂ ∂ x j Γ ℓ i k − ∂ ∂ x k Γ ℓ i j {\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}} == Conformal change == Let g {\displaystyle g} be a Riemannian or pseudo-Riemannian metric on a smooth manifold M {\displaystyle M} , and φ {\displaystyle \varphi } a smooth real-valued function on M {\displaystyle M} . Then g ~ = e 2 φ g {\displaystyle {\tilde {g}}=e^{2\varphi }g} is also a Riemannian metric on M {\displaystyle M} . We say that g ~ {\displaystyle {\tilde {g}}} is (pointwise) conformal to g {\displaystyle g} . Evidently, conformality of metrics is an equivalence relation. Here


id: list-of-formulas-in-riemannian-geometry-20260505-4
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-4.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
=== Hodge operator on p-forms === ∗ ~ i 1 ⋯ i n − p j 1 ⋯ j p = e ( n − 2 p ) φ ∗ i 1 ⋯ i n − p j 1 ⋯ j p {\displaystyle {\widetilde {\ast }}_{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}=e^{(n-2p)\varphi }\ast _{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}} ∗ ~ = e ( n − 2 p ) φ ∗ {\displaystyle {\widetilde {\ast }}=e^{(n-2p)\varphi }\ast } === Codifferential on p-forms === d ∗ ~ j 1 ⋯ j p − 1 i 1 ⋯ i p = e − 2 φ ( d ∗ ) j 1 ⋯ j p − 1 i 1 ⋯ i p − ( n − 2 p ) e − 2 φ ∇ i 1 φ δ j 1 i 2 ⋯ δ j p − 1 i p {\displaystyle {\widetilde {d^{\ast }}}_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}=e^{-2\varphi }(d^{\ast })_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}-(n-2p)e^{-2\varphi }\nabla ^{i_{1}}\varphi \delta _{j_{1}}^{i_{2}}\cdots \delta _{j_{p-1}}^{i_{p}}} d ∗ ~ = e − 2 φ d ∗ − ( n − 2 p ) e − 2 φ ι ∇ φ {\displaystyle {\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }} === Laplacian on functions === Δ ~ Φ = e −


id: list-of-formulas-in-riemannian-geometry-20260505-5
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-5.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
∂ ∂ t R i k = 1 2 ( ∇ p ∇ k v i p + ∇ i ( div ⁡ v ) k − ∇ i ∇ k ( tr g ⁡ v ) − Δ v i k ) + 1 2 R i p v p k − 1 2 R i p k q v p q {\displaystyle {\frac {\partial }{\partial t}}R_{ik}={\frac {1}{2}}{\Big (}\nabla ^{p}\nabla _{k}v_{ip}+\nabla _{i}(\operatorname {div} v)_{k}-\nabla _{i}\nabla _{k}(\operatorname {tr} _{g}v)-\Delta v_{ik}{\Big )}+{\frac {1}{2}}R_{i}^{p}v_{pk}-{\frac {1}{2}}R_{i}{}^{p}{}_{k}{}^{q}v_{pq}} ∂ ∂ t R = div g ⁡ div g ⁡ v − Δ ( tr g ⁡ v ) − ⟨ v , Ric ⟩ g {\displaystyle {\frac {\partial }{\partial t}}R=\operatorname {div} _{g}\operatorname {div} _{g}v-\Delta (\operatorname {tr} _{g}v)-\langle v,\operatorname {Ric} \rangle _{g}} ∂ ∂ t d μ g = 1 2 g p q v p q d μ g {\displaystyle {\frac {\partial }{\partial t}}d\mu _{g}={\frac {1}{2}}g^{pq}v_{pq}\,d\mu _{g}} ∂ ∂ t ∇ i ∇ j Φ = ∇ i ∇ j ∂ Φ ∂ t − 1 2 g k p ( ∇ i v j p + ∇ j v i p − ∇ p v i j ) ∂ Φ ∂ x k {\displaystyle


id: list-of-rules-of-inference-20260505-0
file: data/en.wikipedia.org/wiki/List_of_rules_of_inference-0.md
category: reference
title: List of rules of inference
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rules_of_inference
date_saved: 2026-05-05T08:16:46.303073+00:00
instance: kb-cron
parent: list-of-rules-of-inference-20260505-0
summary:
This is a list of rules of inference, logical laws that relate to mathematical formulae. == Introduction == Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation φ ⊢ ψ {\displaystyle \varphi \vdash \psi } indicates such a subderivation from the temporary assumption φ {\displaystyle \varphi } to ψ {\displaystyle \psi } . == Rules for propositional calculus == === Rules for negations === Reductio ad absurdum (or Negation Introduction) φ ⊢ ψ {\displaystyle \varphi \vdash \psi } φ ⊢ ¬ ψ _ {\displaystyle {\underline {\varphi \vdash \lnot \psi }}} ¬ φ {\displaystyle \lnot \varphi } Reductio ad absurdum (related to the law of excluded middle) ¬ φ ⊢ ψ {\displaystyle \lnot \varphi \vdash \psi


id: list-of-rules-of-inference-20260505-1
file: data/en.wikipedia.org/wiki/List_of_rules_of_inference-1.md
category: reference
title: List of rules of inference
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rules_of_inference
date_saved: 2026-05-05T08:16:46.303073+00:00
instance: kb-cron
parent: list-of-rules-of-inference-20260505-0
summary:
ψ {\displaystyle \psi } Restriction 1: β {\displaystyle \beta } is a variable which does not occur in φ {\displaystyle \varphi } . Restriction 2: There is no occurrence, free or bound, of β {\displaystyle \beta } in ψ {\displaystyle \psi } . Restriction 3: β {\displaystyle \beta } is not mentioned in any hypothesis or undischarged assumptions. == Rules of substructural logic == The following are special cases of universal generalization and existential elimination; these occur in substructural logics, such as linear logic. Rule of weakening (or monotonicity of entailment) (aka no-cloning theorem) α ⊢ β {\displaystyle \alpha \vdash \beta } α , α ⊢ β ¯ {\displaystyle {\overline {\alpha ,\alpha \vdash \beta }}} Rule of contraction (or idempotency of entailment) (aka no-deleting theorem) α , α , γ ⊢ β _ {\displaystyle {\underline {\alpha ,\alpha ,\gamma \vdash \beta }}} α , γ ⊢ β {\displaystyle \alpha ,\gamma \vdash \beta } == Table: Rules of Inference == The rules above can be summed up in the following table. The "Tautology" column shows how to interpret the notation of a given rule. All rules use the basic logic operators. A complete table of "logic operators" is shown by a truth


id: list-of-runge-kutta-methods-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-0.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation d y d t = f ( t , y ) . {\displaystyle {\frac {dy}{dt}}=f(t,y).} Explicit Runge–Kutta methods take the form y n + 1 = y n + h ∑ i = 1 s b i k i k 1 = f ( t n , y n ) , k 2 = f ( t n + c 2 h , y n + h ( a 21 k 1 ) ) , k 3 = f ( t n + c 3 h , y n + h ( a 31 k 1 + a 32 k 2 ) ) , ⋮ k i = f ( t n + c i h , y n + h ∑ j = 1 i − 1 a i j k j ) . {\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\;\;\vdots \\k_{i}&=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}\right).\end{aligned}}} Stages for implicit methods of s stages take the more general form, with the solution to be found over all s k i = f ( t n + c i h , y n + h ∑ j = 1 s a i j k j


id: list-of-runge-kutta-methods-20260505-1
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-1.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
0 0 0 0 0 1 / 3 1 / 3 0 0 0 2 / 3 − 1 / 3 1 0 0 1 1 − 1 1 0 1 / 8 3 / 8 3 / 8 1 / 8 {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/3&1/3&0&0&0\\2/3&-1/3&1&0&0\\1&1&-1&1&0\\\hline &1/8&3/8&3/8&1/8\\\end{array}}} ==== Ralston's fourth-order method ==== This fourth order method has minimum truncation error. 0 0 0 0 0 2 5 2 5 0 0 0 14 − 3 5 16 − 2 889 + 1 428 5 1 024 3 785 − 1 620 5 1 024 0 0 1 − 3 365 + 2 094 5 6 040 − 975 − 3 046 5 2 552 467 040 + 203 968 5 240 845 0 263 + 24 5 1 812 125 − 1000 5 3 828 3 426 304 + 1 661 952 5 5 924 787 30 − 4 5 123 {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\{\frac {2}{5}}&{\frac {2}{5}}&0&0&0\\{\frac {14-3{\sqrt {5}}}{16}}&{\frac {-2\,889+1\,428{\sqrt {5}}}{1\,024}}&{\frac {3\,785-1\,620{\sqrt {5}}}{1\,024}}&0&0\\1&{\frac {-3\,365+2\,094{\sqrt {5}}}{6\,040}}&{\frac {-975-3\,046{\sqrt {5}}}{2\,552}}&{\frac {467\,040+203\,968{\sqrt {5}}}{240\,845}}&0\\\hline &{\frac {263+24{\sqrt {5}}}{1\,812}}&{\frac {125-1000{\sqrt {5}}}{3\,828}}&{\frac {3\,426\,304+1\,661\,952{\sqrt {5}}}{5\,924\,787}}&{\frac {30-4{\sqrt {5}}}{123}}\\\end{array}}} === Fifth-order methods === ==== Nyström's fifth-order method ==== This fifth-order method was a correction of the one proposed originally by Kutta's work. 0 0 0


id: list-of-runge-kutta-methods-20260505-2
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-2.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
=== Gauss–Legendre methods === These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau: 1 2 − 3 6 1 4 1 4 − 3 6 1 2 + 3 6 1 4 + 3 6 1 4 1 2 1 2 1 2 + 3 2 1 2 − 3 2 {\displaystyle {\begin{array}{c|cc}{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}&{\frac {1}{4}}-{\frac {\sqrt {3}}{6}}\\{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}\\\end{array}}} The Gauss–Legendre method of order six has Butcher tableau: 1 2 − 15 10 5 36 2 9 − 15 15 5 36 − 15 30 1 2 5 36 + 15 24 2 9 5 36 − 15 24 1 2 + 15 10 5 36 + 15 30 2 9 + 15 15 5 36 5 18 4 9 5 18 − 5 6 8 3 − 5 6 {\displaystyle {\begin{array}{c|ccc}{\frac {1}{2}}-{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}&{\frac {2}{9}}-{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{30}}\\{\frac {1}{2}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{24}}&{\frac {2}{9}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{24}}\\{\frac {1}{2}}+{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{30}}&{\frac {2}{9}}+{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}\\\hline &{\frac {5}{18}}&{\frac {4}{9}}&{\frac {5}{18}}\\&-{\frac {5}{6}}&{\frac {8}{3}}&-{\frac {5}{6}}\end{array}}} === Diagonally Implicit Runge–Kutta methods ===


id: list-of-runge-kutta-methods-20260505-3
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-3.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
1 / 2 1 / 2 0 0 0 2 / 3 1 / 6 1 / 2 0 0 1 / 2 − 1 / 2 1 / 2 1 / 2 0 1 3 / 2 − 3 / 2 1 / 2 1 / 2 3 / 2 − 3 / 2 1 / 2 1 / 2 {\displaystyle {\begin{array}{c|cccc}1/2&1/2&0&0&0\\2/3&1/6&1/2&0&0\\1/2&-1/2&1/2&1/2&0\\1&3/2&-3/2&1/2&1/2\\\hline &3/2&-3/2&1/2&1/2\\\end{array}}} === Lobatto methods === There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta. ==== Lobatto IIIA methods ==== The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule: 0 0 0 1 1


id: list-of-runge-kutta-methods-20260505-4
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-4.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
The fifth-order method is given by 0 1 9 − 1 − 6 18 − 1 + 6 18 3 5 − 6 10 1 9 11 45 + 7 6 360 11 45 − 43 6 360 3 5 + 6 10 1 9 11 45 + 43 6 360 11 45 − 7 6 360 1 9 4 9 + 6 36 4 9 − 6 36 {\displaystyle {\begin{array}{c|ccc}0&{\frac {1}{9}}&{\frac {-1-{\sqrt {6}}}{18}}&{\frac {-1+{\sqrt {6}}}{18}}\\{\frac {3}{5}}-{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {7{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {43{\sqrt {6}}}{360}}\\{\frac {3}{5}}+{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {43{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {7{\sqrt {6}}}{360}}\\\hline &{\frac {1}{9}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}\\\end{array}}} ==== Radau IIA methods ==== The ci of this method are zeros of d s − 1 d x s − 1 ( x s − 1 ( x − 1 ) s ) {\displaystyle {\frac {d^{s-1}}{dx^{s-1}}}(x^{s-1}(x-1)^{s})} . The first-order method is equivalent to the backward Euler method. The third-order method is given by 1 / 3 5 / 12 − 1 / 12 1 3 / 4 1 / 4 3 / 4 1 / 4 {\displaystyle {\begin{array}{c|cc}1/3&5/12&-1/12\\1&3/4&1/4\\\hline &3/4&1/4\\\end{array}}} The fifth-order method is given by 2 5 − 6 10 11 45 − 7 6


id: list-of-self-intersecting-polygons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_self-intersecting_polygons-0.md
category: reference
title: List of self-intersecting polygons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_self-intersecting_polygons
date_saved: 2026-05-05T08:16:48.847957+00:00
instance: kb-cron
parent: list-of-self-intersecting-polygons-20260505-0
summary:
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: the crossed quadrilateral, with four edges the antiparallelogram, a crossed quadrilateral with alternate edges of equal length the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel Star polygons pentagram, with five edges hexagram, with six edges heptagram, with seven edges octagram, with eight edges enneagram or nonagram, with nine edges decagram, with ten edges hendecagram, with eleven edges dodecagram, with twelve edges icositetragram, with twenty four edges 257-gram, with two hundred and fifty seven edges == See also == List of regular polytopes and compounds § Stars Complex polygon


id: list-of-mathematical-series-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-0.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, 0 0 {\displaystyle 0^{0}} is taken to have the value 1 {\displaystyle 1} { x } {\displaystyle \{x\}} denotes the fractional part of x {\displaystyle x} B n ( x ) {\displaystyle B_{n}(x)} is a Bernoulli polynomial. B n {\displaystyle B_{n}} is a Bernoulli number, and here, B 1 = − 1 2 . {\displaystyle B_{1}=-{\frac {1}{2}}.} E n {\displaystyle E_{n}} is an Euler number. ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. ψ n ( z ) {\displaystyle \psi _{n}(z)} is a polygamma function. Li s ⁡ ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is a polylogarithm. ( n k ) {\displaystyle n \choose k} is binomial coefficient exp ⁡ ( x ) {\displaystyle \exp(x)} denotes exponential of x {\displaystyle x} == Sums of powers == See Faulhaber's formula. ∑ k = 0 m k n − 1 = B n ( m + 1 ) − B n n {\displaystyle \sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}} The first few values are:


id: list-of-mathematical-series-20260505-1
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-1.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 0 ∞ k 2 z k k ! = ( z + z 2 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{2}{\frac {z^{k}}{k!}}=(z+z^{2})e^{z}} (cf. second moment of Poisson distribution) ∑ k = 0 ∞ k 3 z k k ! = ( z + 3 z 2 + z 3 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}} ∑ k = 0 ∞ k 4 z k k ! = ( z + 7 z 2 + 6 z 3 + z 4 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}} ∑ k = 0 ∞ k n z k k ! = z d d z ∑ k = 0 ∞ k n − 1 z k k ! = e z T n ( z ) {\displaystyle \sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)} where T n ( z ) {\displaystyle T_{n}(z)} is the Touchard polynomials. === Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship === ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! = sin ⁡ z {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\sin z} ∑ k = 0 ∞ z


id: list-of-mathematical-series-20260505-2
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-2.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 0 ∞ ( 4 k ) ! 2 4 k 2 ( 2 k ) ! ( 2 k + 1 ) ! z k = 1 − 1 − z z , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!}{2^{4k}{\sqrt {2}}(2k)!(2k+1)!}}z^{k}={\sqrt {\frac {1-{\sqrt {1-z}}}{z}}},|z|<1} ∑ k = 0 ∞ 2 2 k ( k ! ) 2 ( k + 1 ) ( 2 k + 1 ) ! z 2 k + 2 = ( arcsin ⁡ z ) 2 , | z | ≤ 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}(k!)^{2}}{(k+1)(2k+1)!}}z^{2k+2}=\left(\arcsin {z}\right)^{2},|z|\leq 1} ∑ n = 0 ∞ ∏ k = 0 n − 1 ( 4 k 2 + α 2 ) ( 2 n ) ! z 2 n + ∑ n = 0 ∞ α ∏ k = 0 n − 1 [ ( 2 k + 1 ) 2 + α 2 ] ( 2 n + 1 ) ! z 2 n + 1 = e α arcsin ⁡ z , | z | ≤ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(4k^{2}+\alpha ^{2})}{(2n)!}}z^{2n}+\sum _{n=0}^{\infty }{\frac {\alpha \prod _{k=0}^{n-1}[(2k+1)^{2}+\alpha ^{2}]}{(2n+1)!}}z^{2n+1}=e^{\alpha \arcsin {z}},|z|\leq 1} === Binomial coefficients === ( 1


id: list-of-mathematical-series-20260505-3
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-3.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ A ∈ P ( E ) c a r d ( A ) = n 2 n − 1 , where E is a finite set, and card( E ) = n {\displaystyle \sum _{A\ \in \ {\mathcal {P}}(E)}card(A)=n2^{n-1}{\text{, where }}E{\text{ is a finite set, and card(}}E{\text{) = n}}} == Trigonometric functions == Sums of sines and cosines arise in Fourier series. ∑ k = 1 ∞ cos ⁡ ( k θ ) k = − 1 2 ln ⁡ ( 2 − 2 cos ⁡ θ ) = − ln ⁡ ( 2 sin ⁡ θ 2 ) , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta )=-\ln \left(2\sin {\frac {\theta }{2}}\right),0<\theta <2\pi } ∑ k = 1 ∞ sin ⁡ ( k θ ) k = π − θ 2 , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi } ∑ k = 1 ∞ ( − 1 ) k − 1 k cos ⁡ ( k θ ) = 1 2 ln ⁡ ( 2 + 2 cos ⁡ θ ) = ln ⁡ ( 2 cos ⁡ θ 2 ) ,


id: list-of-mathematical-series-20260505-4
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-4.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ m ≠ 0 n − 1 1 1 − z m = n − 1 2 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {1}{1-z_{m}}}={\frac {n-1}{2}}} ∑ m ≠ 0 n − 1 1 ( 1 − z m ) 2 = ( n − 1 ) ( 5 − n ) 12 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {1}{(1-z_{m})^{2}}}={\frac {(n-1)(5-n)}{12}}} Let k {\displaystyle k} be an integer 0 < k < n {\displaystyle 0<k<n} then we also got: ∑ m = 0 n − 1 z m k = 0 {\displaystyle \sum _{m=0}^{n-1}z_{m}^{k}=0} ∑ m = 0 n − 1 ( x − z m ) k = n ⋅ x k {\displaystyle \sum _{m=0}^{n-1}(x-z_{m})^{k}=n\cdot x^{k}} ∑ m ≠ 0 n − 1 ( z m ) k 1 − z m = 2 k − n − 1 2 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {(z_{m})^{k}}{1-z_{m}}}={\frac {2k-n-1}{2}}} ∑ m ≠ 0 n − 1 ( z m ) k ( 1 − z m ) 2 = 6 k ( n + 2 − k ) − ( n + 1 ) ( n + 5 ) 12 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {(z_{m})^{k}}{(1-z_{m})^{2}}}={\frac {6k(n+2-k)-(n+1)(n+5)}{12}}} == Rational functions == ∑ n = a + 1 ∞


id: list-of-mathematical-series-20260505-5
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-5.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
S ( a , b ) = ( − 1 ) c ( ∑ k = 0 ∞ ( − 1 ) k a k + d − ∑ k = 0 c − 1 ( − 1 ) k a k + d ) = ( − 1 ) c ( S ( a , d ) − ∑ k = 0 c − 1 ( − 1 ) k a k + d ) {\displaystyle S(a,b)=(-1)^{c}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak+d}}-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)=(-1)^{c}\left(S(a,d)-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)} what that means is that all the infinite choices of a {\displaystyle a} and b {\displaystyle b} can essentially be boiled down to the cases where g c d ( a , b ) = 1 {\displaystyle gcd(a,b)=1} and a > b > 0 {\displaystyle a>b>0} . If we assume those two things we can then write: S ( a , b ) = 1 a ( π 2 sin ⁡ ( π b a ) − 2 ∑ m = 0 ⌊ a 2 ⌋ cos ⁡ ( π ( 2 m + 1 ) b a ) ln ⁡ ( sin ⁡ ( π 2 m + 1 2 a ) ) )


id: list-of-mathematical-series-20260505-6
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-6.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 1 ∞ 1 T e k = 1 1 + 1 4 + 1 10 + 1 20 + 1 35 + ⋯ = 3 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{Te_{k}}}={\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{20}}+{\frac {1}{35}}+\cdots ={\frac {3}{2}}} Where T e n = ∑ k = 1 n T k {\displaystyle Te_{n}=\sum _{k=1}^{n}T_{k}} === Exponential and logarithms === ∑ k = 0 ∞ 1 ( 2 k + 1 ) ( 2 k + 2 ) = 1 1 × 2 + 1 3 × 4 + 1 5 × 6 + 1 7 × 8 + 1 9 × 10 + ⋯ = ln ⁡ 2 {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k+1)(2k+2)}}={\frac {1}{1\times 2}}+{\frac {1}{3\times 4}}+{\frac {1}{5\times 6}}+{\frac {1}{7\times 8}}+{\frac {1}{9\times 10}}+\cdots =\ln 2} ∑ k = 1 ∞ 1 2 k k = 1 2 + 1 8 + 1 24 + 1 64 + 1 160 + ⋯ = ln ⁡ 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2^{k}k}}={\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{24}}+{\frac {1}{64}}+{\frac {1}{160}}+\cdots =\ln 2} ∑ k = 1 ∞ ( − 1 ) k + 1 2 k k + ∑ k = 1 ∞ ( − 1 ) k + 1 3 k k


id: list-of-set-identities-and-relations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-0.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. The binary operations of set union ( ∪ {\displaystyle \cup } ) and intersection ( ∩ {\displaystyle \cap } ) satisfy many identities. Several of these identities or "laws" have well established names. == Notation == Throughout this article, capital letters (such as A , B , C , L , M , R , S , {\displaystyle A,B,C,L,M,R,S,} and X {\displaystyle X} ) will denote sets. On the left hand side of an identity, typically, L {\displaystyle L} will be the leftmost set, M {\displaystyle M} will be the middle set, and R {\displaystyle R} will be the rightmost set. This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity. For example, the identity ( L ∖ M ) ∖ R = ( L ∖ R ) ∖ ( M ∖ R ) {\displaystyle (L\,\setminus \,M)\,\setminus \,R~=~(L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)} may be read as: ( Left set


id: list-of-set-identities-and-relations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-1.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∪ L = L (Idempotence) L ∩ L = L (Idempotence) L △ L = ∅ (Nilpotence of index 2) L ∖ L = ∅ (Nilpotence of index 2) {\displaystyle {\begin{alignedat}{10}L\cup L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\cap L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\,\triangle \,L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]L\setminus L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]\end{alignedat}}} Domination/Absorbing element: Definition: z {\displaystyle z} is called a left absorbing element of a binary operator ∗ {\displaystyle \,\ast \,} if z ∗ R = z {\displaystyle z\,\ast \,R=z} for all R {\displaystyle R} and it is called a right absorbing element of ∗ {\displaystyle \,\ast \,} if L ∗ z = z {\displaystyle L\,\ast \,z=z} for all L . {\displaystyle L.} A left absorbing element that is also a right absorbing element if called an absorbing element. Absorbing elements are also sometime called annihilating elements or zero elements. A universe set is an absorbing element of binary union ∪ . {\displaystyle \cup .} The empty set ∅ {\displaystyle \varnothing } is an absorbing element of binary intersection ∩ {\displaystyle \cap } and binary Cartesian product × , {\displaystyle \times ,} and it is also a left absorbing element of set subtraction ∖ : {\displaystyle \,\setminus :} X


id: list-of-set-identities-and-relations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-2.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
X ∖ ( L ∩ R ) = ( X ∖ L ) ∪ ( X ∖ R ) Also written ( L ∩ R ) ∁ = L ∁ ∪ R ∁ (De Morgan's law) X ∖ ( L ∪ R ) = ( X ∖ L ) ∩ ( X ∖ R ) Also written ( L ∪ R ) ∁ = L ∁ ∩ R ∁ (De Morgan's law) {\displaystyle {\begin{alignedat}{10}X\setminus (L\cap R)&=(X\setminus L)\cup (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cap R)^{\complement }=L^{\complement }\cup R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]X\setminus (L\cup R)&=(X\setminus L)\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cup R)^{\complement }=L^{\complement }\cap R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]\end{alignedat}}} === Commutativity === Unions, intersection, and symmetric difference are commutative operations: L ∪ R = R ∪ L (Commutativity) L ∩ R = R ∩ L (Commutativity) L △ R = R △ L (Commutativity) {\displaystyle {\begin{alignedat}{10}L\cup R&\;=\;&&R\cup L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\cap R&\;=\;&&R\cap L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\,\triangle R&\;=\;&&R\,\triangle L&&\quad {\text{ (Commutativity)}}\\[1.4ex]\end{alignedat}}} Set subtraction is not commutative. However, the commutativity of set subtraction can be characterized: from ( L ∖ R ) ∩ ( R ∖ L ) = ∅ {\displaystyle (L\,\setminus \,R)\cap (R\,\setminus \,L)=\varnothing } it follows that: L ∖


id: list-of-set-identities-and-relations-20260505-3
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-3.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
===== Empty set ===== A set L {\displaystyle L} is empty if the sentence ∀ x ( x ∉ L ) {\displaystyle \forall x(x\not \in L)} is true, where the notation x ∉ L {\displaystyle x\not \in L} is shorthand for ¬ ( x ∈ L ) . {\displaystyle \lnot (x\in L).} If L {\displaystyle L} is any set then the following are equivalent: L {\displaystyle L} is not empty, meaning that the sentence ¬ [ ∀ x ( x ∉ L ) ] {\displaystyle \lnot [\forall x(x\not \in L)]} is true (literally, the logical negation of " L {\displaystyle L} is empty" holds true). (In classical mathematics) L {\displaystyle L} is inhabited, meaning: ∃ x ( x ∈ L ) {\displaystyle \exists x(x\in L)} In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set L {\displaystyle L} that is not empty (where by definition, " L {\displaystyle L} is empty" means that the statement ∀ x ( x ∉ L ) {\displaystyle \forall x(x\not \in L)} is true) might not have an inhabitant (which is an x {\displaystyle x}


id: list-of-set-identities-and-relations-20260505-4
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-4.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
=== Precedence rules === There is no universal agreement on the order of precedence of the basic set operators. Nevertheless, many authors use precedence rules for set operators, although these rules vary with the author. One common convention is to associate intersection L ∩ R = { x : ( x ∈ L ) ∧ ( x ∈ R ) } {\displaystyle L\cap R=\{x:(x\in L)\land (x\in R)\}} with logical conjunction (and) L ∧ R {\displaystyle L\land R} and associate union L ∪ R = { x : ( x ∈ L ) ∨ ( x ∈ R ) } {\displaystyle L\cup R=\{x:(x\in L)\lor (x\in R)\}} with logical disjunction (or) L ∨ R , {\displaystyle L\lor R,} and then transfer the precedence of these logical operators (where ∧ {\displaystyle \,\land \,} has precedence over ∨ {\displaystyle \,\lor \,} ) to these set operators, thereby giving ∩ {\displaystyle \,\cap \,} precedence over ∪ . {\displaystyle \,\cup .\,} So for example, L ∪ M ∩ R {\displaystyle L\cup M\cap R} would mean L ∪ ( M ∩ R ) {\displaystyle L\cup (M\cap R)} since it would be associated with the logical statement L ∨ M ∧ R = L ∨ ( M ∧


id: list-of-set-identities-and-relations-20260505-5
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-5.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) ∖ R = L ∖ ( M ∖ R ) {\textstyle \;(L\setminus M)\setminus R=L\setminus (M\setminus R)\;} if and only if ( R ∖ M ) ∖ L = R ∖ ( M ∖ L ) , {\displaystyle \;(R\setminus M)\setminus L=R\setminus (M\setminus L),\;} where the only difference between the left and right hand side set equalities is that the locations of L and R {\displaystyle L{\text{ and }}R} have been swapped. === Distributivity === Definition: If ∗ and ∙ {\displaystyle \ast {\text{ and }}\bullet } are binary operators then ∗ {\displaystyle \,\ast \,} left distributes over ∙ {\displaystyle \,\bullet \,} if L ∗ ( M ∙ R ) = ( L ∗ M ) ∙ ( L ∗ R ) for all L , M , R {\displaystyle L\,\ast \,(M\,\bullet \,R)~=~(L\,\ast \,M)\,\bullet \,(L\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R} while ∗ {\displaystyle \,\ast \,} right distributes over ∙ {\displaystyle \,\bullet \,} if ( L ∙ M ) ∗ R = ( L ∗ R ) ∙ ( M ∗ R ) for all L , M , R . {\displaystyle (L\,\bullet \,M)\,\ast \,R~=~(L\,\ast \,R)\,\bullet \,(M\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R.} The operator ∗ {\displaystyle \,\ast


id: list-of-set-identities-and-relations-20260505-6
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-6.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∪ ( M ∩ R ) = ( L ∪ M ) ∩ ( L ∪ R ) (Left-distributivity of ∪ over ∩ ) L ∪ ( M ∪ R ) = ( L ∪ M ) ∪ ( L ∪ R ) (Left-distributivity of ∪ over ∪ ) L ∩ ( M ∪ R ) = ( L ∩ M ) ∪ ( L ∩ R ) (Left-distributivity of ∩ over ∪ ) L ∩ ( M ∩ R ) = ( L ∩ M ) ∩ ( L ∩ R ) (Left-distributivity of ∩ over ∩ ) L ∩ ( M △ R ) = ( L ∩ M ) △ ( L ∩ R ) (Left-distributivity of ∩ over △ ) L × ( M ∩ R ) = ( L × M ) ∩ ( L × R ) (Left-distributivity of × over ∩ ) L × ( M ∪ R ) = ( L × M ) ∪ ( L × R ) (Left-distributivity of × over ∪ ) L × ( M ∖ R ) = ( L × M ) ∖ ( L × R ) (Left-distributivity of × over ∖ )


id: list-of-set-identities-and-relations-20260505-7
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-7.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∖ ( M ∪ R ) ⊆ ( L ∖ M ) ∪ ( L ∖ R ) = L ∖ ( M ∩ R ) {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cap \,R)\\[1.4ex]\end{alignedat}}} L ∖ ( M ∩ R ) ⊇ ( L ∖ M ) ∩ ( L ∖ R ) = L ∖ ( M ∪ R ) {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cap \,R)~&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)\\[1.4ex]\end{alignedat}}} where equality holds for one (or equivalently, for both) of the above two inclusion formulas if and only if L ∩ M = L ∩ R . {\displaystyle L\,\cap \,M=L\,\cap \,R.} The following statements are equivalent: L ∩ M = L ∩ R {\displaystyle L\cap M\,=\,L\cap R} L ∖ M = L ∖ R {\displaystyle L\,\setminus \,M\,=\,L\,\setminus \,R} L ∖ ( M ∩ R ) = ( L ∖ M ) ∩ ( L ∖ R ) ; {\displaystyle L\,\setminus \,(M\,\cap \,R)=(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R);} that is, ∖ {\displaystyle \,\setminus \,} left distributes over ∩ {\displaystyle \,\cap \,} for these three particular sets L ∖ ( M ∪ R ) = ( L ∖ M ) ∪ ( L ∖ R )


id: list-of-set-identities-and-relations-20260505-8
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-8.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) △ R = ( L ∖ ( M ∪ R ) ) ∪ ( R ∖ L ) ∪ ( L ∩ M ∩ R ) (the three outermost sets are pairwise disjoint) {\displaystyle {\begin{alignedat}{4}(L\setminus M)~\triangle ~R&=(L\setminus (M\cup R))\cup (R\setminus L)\cup (L\cap M\cap R)~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}} ( L ∖ M ) × R = ( L × R ) ∖ ( M × R ) (Distributivity) {\displaystyle (L\,\setminus M)\times R=(L\times R)\,\setminus (M\times R)~~~~~{\text{ (Distributivity)}}} ==== L\(M ⁎ R) ==== Set subtraction on the left, and parentheses on the right L ∖ ( M ∪ R ) = ( L ∖ M ) ∩ ( L ∖ R ) (De Morgan's law) = ( L ∖ M ) ∖ R = ( L ∖ R ) ∖ M {\displaystyle {\begin{alignedat}{3}L\setminus (M\cup R)&=(L\setminus M)&&\,\cap \,(&&L\setminus R)~~~~{\text{ (De Morgan's law) }}\\&=(L\setminus M)&&\,\,\setminus &&R\\&=(L\setminus R)&&\,\,\setminus &&M\\\end{alignedat}}} L ∖ ( M ∩ R ) = ( L ∖ M ) ∪ ( L ∖ R ) (De Morgan's law) {\displaystyle {\begin{alignedat}{4}L\setminus (M\cap R)&=(L\setminus M)\cup (L\setminus R)~~~~{\text{ (De Morgan's law) }}\\\end{alignedat}}} where the above two sets that are the subjects of De Morgan's laws always


id: list-of-set-identities-and-relations-20260505-9
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-9.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∪ M ) ∪ ( M ∪ R ) = L ∪ M ∪ R ( L ∪ M ) ∩ ( M ∪ R ) = M ∪ ( L ∩ R ) ( L ∪ M ) ∖ ( M ∪ R ) = L ∖ ( M ∪ R ) ( L ∪ M ) △ ( M ∪ R ) = ( L ∖ ( M ∪ R ) ) ∪ ( R ∖ ( L ∪ M ) ) = ( L △ R ) ∖ M ( L ∩ M ) ∪ ( M ∩ R ) = M ∩ ( L ∪ R ) ( L ∩ M ) ∩ ( M ∩ R ) = L ∩ M ∩ R ( L ∩ M ) ∖ ( M ∩ R ) = ( L ∩ M ) ∖ R ( L ∩ M ) △ ( M ∩ R ) = [ ( L ∩ M ) ∪ ( M ∩ R ) ] ∖ ( L ∩ M ∩ R ) ( L ∖ M ) ∪ ( M ∖ R ) = ( L ∪ M ) ∖


id: list-of-set-identities-and-relations-20260505-10
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-10.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) ∪ ( L ∖ R ) = L ∖ ( M ∩ R ) ( L ∖ M ) ∩ ( L ∖ R ) = L ∖ ( M ∪ R ) ( L ∖ M ) ∖ ( L ∖ R ) = ( L ∩ R ) ∖ M ( L ∖ M ) △ ( L ∖ R ) = L ∩ ( M △ R ) = ( L ∩ M ) △ ( L ∩ R ) {\displaystyle {\begin{alignedat}{9}(L\,\setminus M)&\,\cup \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cup \,R)\\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&L\,\setminus R)&&\;=\;&&(L\,\cap \,R)\,\setminus \,M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&L\,\setminus R)&&\;=\;&&L\,\cap \,(M\,\triangle \,R)\\[1.4ex]&\,&&\,&&\,&&\;=\;&&(L\cap M)\,\triangle \,(L\cap R)\\[1.4ex]\end{alignedat}}} === Other simplifications === Other properties: L ∩ M = R and L ∩ R = M if and only if M = R ⊆ L . {\displaystyle L\cap M=R\;{\text{ and }}\;L\cap R=M\qquad {\text{ if and only if }}\qquad M=R\subseteq L.} If L ⊆ M {\displaystyle L\subseteq M} then L ∖ R = L ∩ ( M ∖ R ) . {\displaystyle L\setminus R=L\cap (M\setminus R).} L × ( M ∖ R ) = ( L × M ) ∖ ( L × R ) {\displaystyle L\times


id: list-of-set-identities-and-relations-20260505-11
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-11.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L × M × R ) ∩ ( L 2 × M 2 × R 2 ) = ( L ∩ L 2 ) × ( M ∩ M 2 ) × ( R ∩ R 2 ) {\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)} === Binary ⋃ of finite ⨯ === ( L × R ) ∪ ( L 2 × R 2 ) = [ ( L ∖ L 2 ) × R ] ∪ [ ( L 2 ∖ L ) × R 2 ] ∪ [ ( L ∩ L 2 ) × ( R ∪ R 2 ) ] = [ L × ( R ∖ R 2 ) ] ∪ [ L 2 × ( R 2 ∖ R ) ] ∪ [ ( L ∪ L 2 ) × ( R ∩ R 2 ) ] {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}} === Difference \ of finite ⨯ === ( L × R ) ∖ ( L 2 × R 2 ) = [ ( L ∖ L 2 )


id: list-of-set-identities-and-relations-20260505-12
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-12.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
== Arbitrary families of sets == Let ( L i ) i ∈ I , {\displaystyle \left(L_{i}\right)_{i\in I},} ( R j ) j ∈ J , {\displaystyle \left(R_{j}\right)_{j\in J},} and ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} be indexed families of sets. Whenever the assumption is needed, then all indexing sets, such as I {\displaystyle I} and J , {\displaystyle J,} are assumed to be non-empty. === Definitions === A family of sets or (more briefly) a family refers to a set whose elements are sets. An indexed family of sets is a function from some set, called its indexing set, into some family of sets. An indexed family of sets will be denoted by ( L i ) i ∈ I , {\displaystyle \left(L_{i}\right)_{i\in I},} where this notation assigns the symbol I {\displaystyle I} for the indexing set and for every index i ∈ I , {\displaystyle i\in I,} assigns the symbol L i {\displaystyle L_{i}} to the value of the function at i . {\displaystyle i.} The function itself may then be denoted by the symbol L ∙ , {\displaystyle L_{\bullet },} which is obtained from


id: list-of-set-identities-and-relations-20260505-13
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-13.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
otherwise, if you are working in a model in which "the class of all things x {\displaystyle x} " is not a set (by far the most common situation) then ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} is undefined because ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} consists of everything, which makes ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} a proper class and not a set. Assumption: Henceforth, whenever a formula requires some indexing set to be non-empty in order for an arbitrary intersection to be well-defined, then this will automatically be assumed without mention. A consequence of this is the following assumption/definition: A finite intersection of sets or an intersection of finitely many sets refers to the intersection of a finite collection of one or more sets. Some authors adopt the so called nullary intersection convention, which is the convention that an empty intersection of sets is equal to some canonical set. In particular, if all sets are subsets of some set X {\displaystyle X} then some author may declare that the empty intersection of these sets be equal to X


id: list-of-set-identities-and-relations-20260505-14
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-14.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
⋂ i ∈ I ( L i ∪ R i ) {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~\color {Red}{\neq }\color {Black}{}~~\bigcap _{i\in I}\left(L_{i}\cup R_{i}\right)~} (an example of this is given above). The single intersection on the right hand side must be over all pairs ( i , j ) ∈ I × I : {\displaystyle (i,j)\in I\times I:} ( ⋂ i ∈ I L i ) ∪ ( ⋂ i ∈ I R i ) = ⋂ j ∈ I i ∈ I , ( L i ∪ R j ) . {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~=~~\bigcap _{\stackrel {i\in I,}{j\in I}}\left(L_{i}\cup R_{j}\right).~} ==== Arbitrary ⋂'s and arbitrary ⋃'s ==== ===== Incorrectly distributing by swapping ⋂ and ⋃ ===== Naively swapping ⋃ i ∈ I {\displaystyle \;{\textstyle \bigcup \limits _{i\in I}}\;} and ⋂ j ∈ J {\displaystyle \;{\textstyle \bigcap \limits _{j\in J}}\;} may produce a different set The following inclusion always holds: In general, equality need not hold and moreover, the right hand side depends on how for each fixed i ∈ I , {\displaystyle i\in I,} the sets ( S i , j ) j ∈ J {\displaystyle \left(S_{i,j}\right)_{j\in J}} are labelled; and analogously, the left hand side


id: list-of-set-identities-and-relations-20260505-15
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-15.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Equality in Inclusion 1 ∪∩ is a subset of ∩∪ can hold under certain circumstances, such as in 7e, which is the special case where ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is ( L i ∖ R j ) ( i , j ) ∈ I × J {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(i,j)\in I\times J}} (that is, S i , j : = L i ∖ R j {\displaystyle S_{i,j}\colon =L_{i}\setminus R_{j}} with the same indexing sets I {\displaystyle I} and J {\displaystyle J} ), or such as in 7f, which is the special case where ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is ( L i ∖ R j ) ( j , i ) ∈ J × I {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(j,i)\in J\times I}} (that is, S ^ j , i : = L i ∖ R j {\displaystyle {\hat {S}}_{j,i}\colon =L_{i}\setminus R_{j}} with the indexing sets I {\displaystyle I} and J {\displaystyle J} swapped). For a correct formula that extends the distributive laws, an approach other than just switching ∪ {\displaystyle \cup } and ∩


id: list-of-set-identities-and-relations-20260505-16
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-16.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Example application: To apply the general formula to the case of ( C k ) k ∈ K {\displaystyle \left(C_{k}\right)_{k\in K}} and ( D l ) l ∈ L , {\displaystyle \left(D_{l}\right)_{l\in L},} use I : = { 1 , 2 } , {\displaystyle I\colon =\{1,2\},} J 1 : = K , {\displaystyle J_{1}\colon =K,} J 2 : = L , {\displaystyle J_{2}\colon =L,} and let T 1 , k : = C k {\displaystyle T_{1,k}\colon =C_{k}} for all k ∈ J 1 {\displaystyle k\in J_{1}} and let T 2 , l : = D l {\displaystyle T_{2,l}\colon =D_{l}} for all l ∈ J 2 . {\displaystyle l\in J_{2}.} Every map f ∈ ∏ J ∙ = def ∏ i ∈ I J i = J 1 × J 2 = K × L {\displaystyle f\in {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}=J_{1}\times J_{2}=K\times L} can be bijectively identified with the pair ( f ( 1 ) , f ( 2 ) ) ∈ K × L {\displaystyle \left(f(1),f(2)\right)\in K\times L} (the inverse sends ( k , l ) ∈ K × L {\displaystyle (k,l)\in K\times L} to the map f ( k , l ) ∈


id: list-of-set-identities-and-relations-20260505-17
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-17.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
and if I = J {\displaystyle I=J} then also: === Cartesian products Π of arbitrarily many sets === ==== Intersections ⋂ of Π ==== If ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is a family of sets then Moreover, a tuple ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} belongs to the set in Eq. 8 above if and only if x i ∈ S i , j {\displaystyle x_{i}\in S_{i,j}} for all i ∈ I {\displaystyle i\in I} and all j ∈ J . {\displaystyle j\in J.} In particular, if ( L i ) i ∈ I {\displaystyle \left(L_{i}\right)_{i\in I}} and ( R i ) i ∈ I {\displaystyle \left(R_{i}\right)_{i\in I}} are two families indexed by the same set then ( ∏ i ∈ I L i ) ∩ ∏ i ∈ I R i = ∏ i ∈ I ( L i ∩ R i ) {\displaystyle \left(\prod _{i\in I}L_{i}\right)\cap \prod _{i\in I}R_{i}~=~\prod _{i\in I}\left(L_{i}\cap R_{i}\right)} So for instance, ( L × R ) ∩ ( L 2 × R 2 ) = ( L ∩ L 2 ) × ( R ∩ R


id: list-of-set-identities-and-relations-20260505-18
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-18.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L × ( ⋃ i ∈ I R i ) = ⋃ i ∈ I ( L × R i ) (Left-distributivity of × over ∪ ) L × ( ⋂ i ∈ I R i ) = ⋂ i ∈ I ( L × R i ) (Left-distributivity of × over ⋂ i ∈ I when I ≠ ∅ ) ( ⋃ i ∈ I L i ) × R = ⋃ i ∈ I ( L i × R ) (Right-distributivity of × over ∪ ) ( ⋂ i ∈ I L i ) × R = ⋂ i ∈ I ( L i × R ) (Right-distributivity of × over ⋂ i ∈ I when I ≠ ∅ ) {\displaystyle {\begin{alignedat}{5}L\times \left(\bigcup _{i\in I}R_{i}\right)&\;=\;\;&&\bigcup _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times \left(\bigcap _{i\in I}R_{i}\right)&\;=\;\;&&\bigcap _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\left(\bigcup _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcup _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]\left(\bigcap _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcap _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\end{alignedat}}} ==== Distributing arbitrary Π over arbitrary ⋃ ==== Suppose that


id: list-of-set-identities-and-relations-20260505-19
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-19.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( ∏ i ∈ I L i ) ∖ ∏ i ∈ I R i = ⋃ j ∈ I ∏ i ∈ I { L j ∖ R j if i = j L i if i ≠ j = ⋃ j ∈ I [ ( L j ∖ R j ) × ∏ j ≠ i i ∈ I , L i ] = ⋃ L j ⊈ R j j ∈ I , [ ( L j ∖ R j ) × ∏ j ≠ i i ∈ I , L i ] {\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\setminus ~\prod _{i\in I}R_{i}~&=~\;~\bigcup _{j\in I}\;~\prod _{i\in I}{\begin{cases}L_{j}\,\setminus \,R_{j}&{\text{ if }}i=j\\L_{i}&{\text{ if }}i\neq j\\\end{cases}}\\[0.5ex]~&=~\;~\bigcup _{j\in I}\;~{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.5ex]~&=~\bigcup _{\stackrel {j\in I,}{L_{j}\not \subseteq R_{j}}}{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.3ex]\end{alignedat}}} so for instance, ( L × R ) ∖ ( L 2 × R 2 ) = [ ( L ∖ L 2 ) × R ] ∪ [ L × ( R ∖ R 2 ) ] {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}} and ( L × M × R ) ∖ ( L 2 × M 2 ×


id: list-of-set-identities-and-relations-20260505-20
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-20.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Any such set R {\displaystyle R} necessarily contains f ( A ) {\displaystyle f(A)} as a subset. Any set not entirely contained in the domain of f {\displaystyle f} cannot be f {\displaystyle f} -saturated. A = f − 1 ( f ( A ) ) . {\displaystyle A=f^{-1}(f(A)).} A ⊇ f − 1 ( f ( A ) ) {\displaystyle A\supseteq f^{-1}(f(A))} and A ⊆ domain ⁡ f . {\displaystyle A\subseteq \operatorname {domain} f.} The inclusion L ∩ domain ⁡ f ⊆ f − 1 ( f ( L ) ) {\displaystyle L\cap \operatorname {domain} f\subseteq f^{-1}(f(L))} always holds, where if A ⊆ domain ⁡ f {\displaystyle A\subseteq \operatorname {domain} f} then this becomes A ⊆ f − 1 ( f ( A ) ) . {\displaystyle A\subseteq f^{-1}(f(A)).} A ⊆ domain ⁡ f {\displaystyle A\subseteq \operatorname {domain} f} and if a ∈ A {\displaystyle a\in A} and x ∈ domain ⁡ f {\displaystyle x\in \operatorname {domain} f} satisfy f ( x ) = f ( a ) , {\displaystyle f(x)=f(a),} then x ∈ A . {\displaystyle x\in A.} Whenever a fiber of f {\displaystyle f} intersects A , {\displaystyle A,} then A {\displaystyle A} contains the entire fiber.


id: list-of-set-identities-and-relations-20260505-21
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-21.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
If ( A i ) i ∈ I {\displaystyle \left(A_{i}\right)_{i\in I}} is a family of arbitrary subsets of X = domain ⁡ f , {\displaystyle X=\operatorname {domain} f,} which means that A i ⊆ X {\displaystyle A_{i}\subseteq X} for all i , {\displaystyle i,} then Conditional Equality 10a becomes: === (Pre)Images of binary set operations === Throughout, let L {\displaystyle L} and R {\displaystyle R} be any sets and let f : X → Y {\displaystyle f:X\to Y} be any function. Summary As the table below shows, set equality is not guaranteed only for images of: intersections, set subtractions, and symmetric differences. Preimages preserve set operations Preimages of sets are well-behaved with respect to all basic set operations: f − 1 ( L ∪ R ) = f − 1 ( L ) ∪ f − 1 ( R ) f − 1 ( L ∩ R ) = f − 1 ( L ) ∩ f − 1 ( R ) f − 1 ( L ∖ R ) = f − 1 ( L ) ∖ f − 1 ( R ) f − 1 ( L △ R ) = f − 1 ( L ) △


id: list-of-set-identities-and-relations-20260505-22
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-22.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
The containment f ( L △ R ) ⊋ f ( L ) △ f ( R ) {\displaystyle ~f(L~\triangle ~R)~\supsetneq ~f(L)~\triangle ~f(R)~} is strict: { y } = f ( L △ R ) ≠ f ( L ) △ f ( R ) = { y } △ { y } = ∅ {\displaystyle \{y\}~=~f\left(L~\triangle ~R\right)~\neq ~f(L)~\triangle ~f(R)~=~\{y\}\triangle \{y\}~=~\varnothing } In words: functions might not distribute over symmetric difference △ {\displaystyle \,\triangle \,} (which can be defined as the set subtraction of two sets: L △ R = ( L ∪ R ) ∖ ( L ∩ R ) {\displaystyle L\triangle R=(L\cup R)\setminus (L\cap R)} ). The containment f ( L ∩ R ) ⊊ f ( L ) ∩ f ( R ) {\displaystyle ~f(L\cap R)~\subsetneq ~f(L)\cap f(R)~} is strict: ∅ = f ( ∅ ) = f ( L ∩ R ) ≠ f ( L ) ∩ f ( R ) = { y } ∩ { y } = { y } {\displaystyle \varnothing ~=~f(\varnothing )~=~f(L\cap R)~\neq ~f(L)\cap f(R)~=~\{y\}\cap \{y\}~=~\{y\}} In words: functions might not distribute over set intersection ∩ {\displaystyle \,\cap \,} (which can be defined as the set subtraction of two sets:


id: list-of-set-identities-and-relations-20260505-23
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-23.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
f ( L △ R ) = f ( L ) △ f ( R ) for all L , R ⊆ X . {\displaystyle f(L\,\triangle \,R)=f(L)\,\triangle \,f(R)\;{\text{ for all }}\,L,R\subseteq X.} (The equals sign = {\displaystyle \,=\,} can be replaced with ⊆ {\displaystyle \,\subseteq \,} ). Any one of the four statements (b) - (e) but with the words "for all" replaced with any one of the following: "for all singleton subsets" In particular, the statement that results from (d) gives a characterization of injectivity that explicitly involves only one point (rather than two): f {\displaystyle f} is injective if and only if f ( x ) ∉ f ( X ∖ { x } ) for every x ∈ X . {\displaystyle f(x)\not \in f(X\setminus \{x\})\;{\text{ for every }}\,x\in X.} "for all disjoint singleton subsets" For statement (d), this is the same as: "for all singleton subsets" (because the definition of "pairwise disjoint" is satisfies vacuously by any family that consists of exactly 1 set). "for all disjoint subsets" In particular, if a map is not known to be injective then barring additional information, there is no guarantee that any of the equalities in statements (b) - (e)


id: list-of-set-identities-and-relations-20260505-24
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-24.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∩ domain ⁡ f ⊆ R {\displaystyle L\cap \operatorname {domain} f\,\subseteq R} R ⊆ L {\displaystyle R\subseteq L} L ⊆ R {\displaystyle L\subseteq R} In addition, the following always hold: f ( f − 1 ( L ) ∩ R ) = L ∩ f ( R ) {\displaystyle f\left(f^{-1}(L)\cap R\right)~=~L\cap f(R)} f ( f − 1 ( L ) ∪ R ) = ( L ∩ Im ⁡ f ) ∪ f ( R ) {\displaystyle f\left(f^{-1}(L)\cup R\right)~=~(L\cap \operatorname {Im} f)\cup f(R)} ===== Conditions for f(L\R) = f(L)\f(R) ===== f ( L ∖ R ) ⊇ f ( L ) ∖ f ( R ) always holds {\displaystyle f(L\setminus R)~\supseteq ~f(L)\setminus f(R)\qquad \qquad {\text{ always holds}}} Characterizations of equality: The following statements are equivalent: f ( L ∖ R ) = f ( L ) ∖ f ( R ) {\displaystyle f(L\setminus R)~=~f(L)\setminus f(R)} f ( L ∖ R ) ⊆ f ( L ) ∖ f ( R ) {\displaystyle f(L\setminus R)~\subseteq ~f(L)\setminus f(R)} L ∩ f − 1 ( f ( R ) ) ⊆ R {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} L ∩ f − 1 ( f ( R ) ) = L ∩ R ∩


id: list-of-set-identities-and-relations-20260505-25
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-25.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Characterizations of equality: The following statements are equivalent: f ( L △ R ) = f ( L ) △ f ( R ) {\displaystyle f\left(L~\triangle ~R\right)=f(L)~\triangle ~f(R)} f ( L △ R ) ⊆ f ( L ) △ f ( R ) {\displaystyle f\left(L~\triangle ~R\right)\subseteq f(L)~\triangle ~f(R)} f ( L ∖ R ) = f ( L ) ∖ f ( R ) {\displaystyle f(L\,\setminus \,R)=f(L)\,\setminus \,f(R)} and f ( R ∖ L ) = f ( R ) ∖ f ( L ) {\displaystyle f(R\,\setminus \,L)=f(R)\,\setminus \,f(L)} f ( L ∖ R ) ⊆ f ( L ) ∖ f ( R ) {\displaystyle f(L\,\setminus \,R)\subseteq f(L)\,\setminus \,f(R)} and f ( R ∖ L ) ⊆ f ( R ) ∖ f ( L ) {\displaystyle f(R\,\setminus \,L)\subseteq f(R)\,\setminus \,f(L)} L ∩ f − 1 ( f ( R ) ) ⊆ R {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} and R ∩ f − 1 ( f ( L ) ) ⊆ L {\displaystyle R\cap f^{-1}(f(L))~\subseteq ~L} The inclusions L ∩ f − 1 ( f ( R ) ) ⊆ f − 1 ( f ( L ) ) {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L))} and R ∩ f −


id: list-of-set-identities-and-relations-20260505-26
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-26.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
f ( X ∖ R ) = f ( X ) ∖ { y ∈ f ( R ) : f − 1 ( y ) ⊆ R } . {\displaystyle f(X\setminus R)\,=\,f(X)\setminus \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}.} From this it follows that: f ( X ∖ R ) = f ( X ) ∖ f ( R ) if and only if f ( R ) = { y ∈ f ( R ) : f − 1 ( y ) ⊆ R } if and only if f − 1 ( f ( R ) ) ⊆ R . {\displaystyle f(X\setminus R)=f(X)\setminus f(R)\quad {\text{ if and only if }}\quad f(R)=\left\{y\in f(R):f^{-1}(y)\subseteq R\right\}\quad {\text{ if and only if }}\quad f^{-1}(f(R))\subseteq R.} If f R := { y ∈ f ( X ) : f − 1 ( y ) ⊆ R } {\displaystyle f_{R}:=\left\{y\in f(X):f^{-1}(y)\subseteq R\right\}} then f ( X ∖ R ) = f ( X ) ∖ f R , {\displaystyle f(X\setminus R)=f(X)\setminus f_{R},} which can be written more symmetrically as f ( X ∖ R ) = f X ∖ f R {\displaystyle f(X\setminus R)=f_{X}\setminus f_{R}} (since f X = f ( X ) {\displaystyle f_{X}=f(X)} ). =====


id: list-of-set-identities-and-relations-20260505-27
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-27.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∩ f − 1 ( y ) ⊆ L ∖ R {\displaystyle L\cap f^{-1}(y)\subseteq L\setminus R~} if and only if L ∩ R ∩ f − 1 ( y ) = ∅ {\displaystyle ~L\cap R\cap f^{-1}(y)=\varnothing ~} if and only if R ∩ f − 1 ( y ) ⊆ R ∖ L {\displaystyle ~R\cap f^{-1}(y)\subseteq R\setminus L~} if and only if y ∉ f ( L ∩ R ) . {\displaystyle ~y\not \in f(L\cap R).} The sets U {\displaystyle U} and V {\displaystyle V} mentioned above could, in particular, be any of the sets f ( L ∪ R ) , Im ⁡ f , {\displaystyle f(L\cup R),\;\operatorname {Im} f,} or Y , {\displaystyle Y,} for example. === (Pre)Images of set operations on (pre)images === Let L {\displaystyle L} and R {\displaystyle R} be arbitrary sets, f : X → Y {\displaystyle f:X\to Y} be any map, and let A ⊆ X {\displaystyle A\subseteq X} and C ⊆ Y . {\displaystyle C\subseteq Y.} (Pre)Images of operations on images Since f ( L ) ∖ f ( L ∖ R ) = { y ∈ f ( L ∩ R ) : L ∩ f − 1 ( y


id: list-of-set-identities-and-relations-20260505-28
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-28.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
The map F ∙ = ( F j ) j ∈ J : X → ∏ j ∈ J Y j {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}} should not be confused with the Cartesian product ∏ j ∈ J F j {\displaystyle \prod _{j\in J}F_{j}} of these maps, which is by definition is the map ∏ j ∈ J F j : ∏ j ∈ J X → ∏ j ∈ J Y j ( x j ) j ∈ J ↦ ( F j ( x j ) ) j ∈ J {\displaystyle {\begin{alignedat}{4}\prod _{j\in J}F_{j}:\;&&\prod _{j\in J}X&&~\;\to \;~&\prod _{j\in J}Y_{j}\\[0.3ex]&&\left(x_{j}\right)_{j\in J}&&~\;\mapsto \;~&\left(F_{j}\left(x_{j}\right)\right)_{j\in J}\\\end{alignedat}}} with domain ∏ j ∈ J X = X J {\displaystyle \prod _{j\in J}X=X^{J}} rather than X . {\displaystyle X.} Preimage and images of a Cartesian product Suppose F ∙ = ( F j ) j ∈ J : X → ∏ j ∈ J Y j . {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}.} If A ⊆ X {\displaystyle A~\subseteq ~X} then F ∙ ( A ) ⊆ ∏ j ∈ J F j ( A ) . {\displaystyle F_{\bullet }(A)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\prod _{j\in J}F_{j}(A).} If B ⊆ ∏ j ∈ J


id: list-of-set-identities-and-relations-20260505-29
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-29.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
where N {\displaystyle \mathbb {N} } denotes the natural numbers. A notation S ∙ = ( S i ) i ∈ I {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}} indicates that S ∙ {\displaystyle S_{\bullet }} is a net directed by ( I , ≤ ) , {\displaystyle (I,\leq ),} which (by definition) is a sequence if the set I , {\displaystyle I,} which is called the net's indexing set, is the natural numbers (that is, if I = N {\displaystyle I=\mathbb {N} } ) and ≤ {\displaystyle \,\leq \,} is the natural order on N . {\displaystyle \mathbb {N} .} Disjoint and monotone sequences of sets If S i ∩ S j = ∅ {\displaystyle S_{i}\cap S_{j}=\varnothing } for all distinct indices i ≠ j {\displaystyle i\neq j} then S ∙ {\displaystyle S_{\bullet }} is called a pairwise disjoint or simply a disjoint. A sequence or net S ∙ {\displaystyle S_{\bullet }} of set is called increasing or non-decreasing if (resp. decreasing or non-increasing) if for all indices i ≤ j , {\displaystyle i\leq j,} S i ⊆ S j {\displaystyle S_{i}\subseteq S_{j}} (resp. S i ⊇ S j {\displaystyle S_{i}\supseteq S_{j}} ). A sequence or net S ∙ {\displaystyle S_{\bullet }}


id: list-of-set-identities-and-relations-20260505-30
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-30.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
closed under finite intersections (resp. closed under finite unions) if whenever L , R ∈ L {\displaystyle L,R\in {\mathcal {L}}} then L ∩ R ∈ L {\displaystyle L\cap R\in {\mathcal {L}}} (respectively, L ∪ R ∈ L {\displaystyle L\cup R\in {\mathcal {L}}} ). closed under countable intersections (resp. closed under countable unions) if whenever L 1 , L 2 , L 3 , … {\displaystyle L_{1},L_{2},L_{3},\ldots } are elements of L {\displaystyle {\mathcal {L}}} then so is their intersections ⋂ i = 1 ∞ L i := L 1 ∩ L 2 ∩ L 3 ∩ ⋯ {\displaystyle \bigcap _{i=1}^{\infty }L_{i}:=L_{1}\cap L_{2}\cap L_{3}\cap \cdots } (resp. so is their union ⋃ i = 1 ∞ L i := L 1 ∪ L 2 ∪ L 3 ∪ ⋯ {\displaystyle \bigcup _{i=1}^{\infty }L_{i}:=L_{1}\cup L_{2}\cup L_{3}\cup \cdots } ). closed under complementation in (or with respect to) X {\displaystyle X} if whenever L ∈ L {\displaystyle L\in {\mathcal {L}}} then X ∖ L ∈ L . {\displaystyle X\setminus L\in {\mathcal {L}}.} A family L {\displaystyle {\mathcal {L}}} of sets is called a/an: π−system if L ≠ ∅ {\displaystyle {\mathcal {L}}\neq \varnothing } and L {\displaystyle {\mathcal {L}}} is closed under finite-intersections. Every


id: list-of-set-identities-and-relations-20260505-31
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-31.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ( ∪ ) { X } = { X } if L ≠ ∅ {\displaystyle {\mathcal {L}}\;(\cup )\;\{X\}=\{X\}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } L ( ∩ ) { ∅ } = { ∅ } if L ≠ ∅ {\displaystyle {\mathcal {L}}\;(\cap )\;\{\varnothing \}=\{\varnothing \}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } L ( ∪ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\cup )\;\varnothing =\varnothing } L ( ∩ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\cap )\;\varnothing =\varnothing } L ( ∖ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\setminus )\;\varnothing =\varnothing } ∅ ( ∖ ) R = ∅ {\displaystyle \varnothing \;(\setminus )\;{\mathcal {R}}=\varnothing } === Power set === P ( L ∩ R ) = P ( L ) ∩ P ( R ) {\displaystyle {\mathcal {P}}(L\cap R)~=~{\mathcal {P}}(L)\cap {\mathcal {P}}(R)} P ( L ∪ R ) = P ( L ) ( ∪ ) P ( R ) ⊇ P ( L ) ∪ P ( R ) . {\displaystyle {\mathcal {P}}(L\cup R)~=~{\mathcal {P}}(L)\ (\cup )\ {\mathcal {P}}(R)~\supseteq ~{\mathcal {P}}(L)\cup {\mathcal {P}}(R).} If L {\displaystyle L} and R {\displaystyle R} are subsets of a vector space X {\displaystyle X} and if s {\displaystyle s} is a scalar then P ( s


id: list-of-set-identities-and-relations-20260505-32
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-32.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
== References == Artin, Michael (1991). Algebra. Prentice Hall. ISBN 81-203-0871-9. Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1-85233-905-5.. Bylinski, Czeslaw (2004). "Some Basic Properties of Sets". Journal of Formalized Mathematics. 1. Retrieved 5 October 2021. Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. "SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS". Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011. Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303. Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020. Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. {{cite book}}: ISBN / Date incompatibility


id: list-of-alternative-set-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_alternative_set_theories-0.md
category: reference
title: List of alternative set theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_alternative_set_theories
date_saved: 2026-05-05T08:16:54.419564+00:00
instance: kb-cron
parent: list-of-alternative-set-theories-20260505-0
summary:
In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. == Alternative set theories == Alternative set theories include: Vopěnka's alternative set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory Pocket set theory Naive set theory S (set theory) Double extension set theory Kripke–Platek set theory Kripke–Platek set theory with urelements Scott–Potter set theory Constructive set theory Zermelo set theory General set theory Mac Lane set theory == See also == Non-well-founded set theory List of first-order theories § Set theories == Notes ==


id: list-of-set-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_set_theory_topics-0.md
category: reference
title: List of set theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_theory_topics
date_saved: 2026-05-05T08:16:55.603082+00:00
instance: kb-cron
parent: list-of-set-theory-topics-20260505-0
summary:
This page is a list of articles related to set theory. == Articles on individual set theory topics == == Lists related to set theory == Glossary of set theory List of large cardinal properties List of properties of sets of reals List of set identities and relations == Set theorists == == Societies and organizations == Association for Symbolic Logic The Cabal


id: list-of-types-of-sets-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_sets-0.md
category: reference
title: List of types of sets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_sets
date_saved: 2026-05-05T08:16:56.774657+00:00
instance: kb-cron
parent: list-of-types-of-sets-20260505-0
summary:
Sets can be classified according to the properties they have. == Relative to set theory == Empty set Finite set, Infinite set Countable set, Uncountable set Power set == Relative to a topology == Closed set Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere dense set == Relative to a metric == Bounded set Totally bounded set == Relative to measurability == Borel set Baire set Measurable set, Non-measurable set Universally measurable set == Relative to a measure == Negligible set Null set Haar null set == In a linear space == Convex set Balanced set, Absolutely convex set == Relative to the real/complex numbers == Fractal set == Ways of defining sets/Relation to descriptive set theory == Recursive set Recursively enumerable set Arithmetical set Diophantine set Hyperarithmetical set Analytical set Analytic set, Coanalytic set Suslin set Projective set Inhabited set == More general objects still called sets == Multiset == See also == List of set identities and relations – Equalities for combinations of sets List of types of functions


id: list-of-shapes-with-known-packing-constant-20260505-0
file: data/en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant-0.md
category: reference
title: List of shapes with known packing constant
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant
date_saved: 2026-05-05T08:16:58.046518+00:00
instance: kb-cron
parent: list-of-shapes-with-known-packing-constant-20260505-0
summary:
The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown. The following is a list of bodies in Euclidean spaces whose packing constant is known. Fejes Tóth proved that in the plane, a point symmetric body has a packing constant that is equal to its translative packing constant and its lattice packing constant. Therefore, any such body for which the lattice packing constant was previously known, such as any ellipse, consequently has a known packing constant. In addition to these bodies, the packing constants of hyperspheres in 8 and 24 dimensions are almost exactly known. == References ==


id: list-of-solids-derived-from-the-sphere-20260505-0
file: data/en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere-0.md
category: reference
title: List of solids derived from the sphere
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere
date_saved: 2026-05-05T08:16:59.236190+00:00
instance: kb-cron
parent: list-of-solids-derived-from-the-sphere-20260505-0
summary:
This page lists solids derived from a sphere. == Solids from cutting a sphere with one or more planes == Dome Spherical cap Spherical sector Spherical segment Spherical shell Spherical wedge == Solids from deforming a sphere == Ellipsoid Spheroid Solid bounded by Morin surface Any Genus 0 surface == Solids from intersecting a sphere with other solids or curved planes == Reuleaux tetrahedron Spherical lens == Notes ==


id: list-of-spherical-symmetry-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_spherical_symmetry_groups-0.md
category: reference
title: List of spherical symmetry groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups
date_saved: 2026-05-05T08:17:01.639486+00:00
instance: kb-cron
parent: list-of-spherical-symmetry-groups-20260505-0
summary:
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway used a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. == Involutional symmetry == There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci). == Cyclic symmetry == There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry) == Dihedral symmetry == There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).


id: list-of-textbooks-in-thermodynamics-and-statistical-mechanics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics-0.md
category: reference
title: List of textbooks in thermodynamics and statistical mechanics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics
date_saved: 2026-05-05T08:17:02.992953+00:00
instance: kb-cron
parent: list-of-textbooks-in-thermodynamics-and-statistical-mechanics-20260505-0
summary:
A list of notable textbooks in thermodynamics and statistical mechanics, arranged by category and date. == Only or mainly thermodynamics == Fermi, Enrico (1956). Thermodynamics (New ed.). Dover Publications. ISBN 978-0486603612. {{cite book}}: ISBN / Date incompatibility (help) Van Ness, H. C. (1983). Understanding Thermodynamics. Dover Publications. ISBN 978-0486632773. Callen, Herbert (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8. Finn, C. B. P. (1993). Thermal Physics (2nd ed.). United States of America: CRC Press. ISBN 0-7487-4379-0. Zemansky, Mark W.; Dittman, Richard (1997). Heat and Thermodynamics: An Intermediate Textbook (7th ed.). McGraw-Hill. ISBN 978-0-070-17059-9. Hanson, Robert M.; Green, Susan (2008). Introduction to Molecular Thermodynamics. University Science Books. ISBN 978-1891389498, 978-1-938787-63-8. Pokrovskii, Vladimir (2020). Thermodynamics of Complex Systems: Principles and applications. IOP Publishing, Bristol, UK. ISBN 978-0-7503-3451-8. == Both thermodynamics and statistical mechanics == Reif, Frederick (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill. ISBN 0-07-051800-9. Sears, Francis W. (1975). Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. Addison Wesley. ISBN 020106894X. Kittel, Charles (1969). Thermal Physics. Chichester: Wiley. ISBN 0-471-49030-X. 2e Kittel, Charles; and Kroemer, Herbert (1980) New York: W.H. Freeman ISBN 0-7167-1088-9 Mandl, Franz (1971). Statistical physics. Chichester: Wiley. ISBN 0-471-56658-6. 2e (1988)


