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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical proof | 2/4 | https://en.wikipedia.org/wiki/Mathematical_proof | reference | science, encyclopedia | 2026-05-05T07:24:53.330614+00:00 | kb-cron |
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two even integers is always even:
Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence, the sum of any two even integers is even. This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property.
=== Proof by mathematical induction ===
Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proven base case), it follows that all (usually infinitely many) cases are provable. This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent, which can be used, for example, to prove the irrationality of the square root of two. A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers: Let N = {1, 2, 3, 4, ...} be the set of natural numbers, and let P(n) be a mathematical statement involving the natural number n belonging to N such that
(i) P(1) is true, i.e., P(n) is true for n = 1. (ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true. Then P(n) is true for all natural numbers n. For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd":
(i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true. (ii) For any n, if 2n − 1 is odd (P(n)), then (2n − 1) + 2 must also be odd, because adding 2 to an odd number results in an odd number. But (2n − 1) + 2 = 2n + 1 = 2(n+1) − 1, so 2(n+1) − 1 is odd (P(n+1)). So P(n) implies P(n+1). Thus 2n − 1 is odd, for all positive integers n. The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".
=== Proof by contraposition ===
Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p". For example, contraposition can be used to establish that, given an integer
x
{\displaystyle x}
, if
x
2
{\displaystyle x^{2}}
is even, then
x
{\displaystyle x}
is even:
Suppose
x
{\displaystyle x}
is not even. Then
x
{\displaystyle x}
is odd. The product of two odd numbers is odd, hence
x
2
=
x
⋅
x
{\displaystyle x^{2}=x\cdot x}
is odd. Thus
x
2
{\displaystyle x^{2}}
is not even. Thus, if
x
2
{\displaystyle x^{2}}
is even, the supposition must be false, so
x
{\displaystyle x}
has to be even.
=== Proof by contradiction ===
In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that
2
{\displaystyle {\sqrt {2}}}
is an irrational number:
Suppose that
2
{\displaystyle {\sqrt {2}}}
were a rational number. Then it could be written in lowest terms as
2
=
a
b
{\displaystyle {\sqrt {2}}={a \over b}}
where a and b are non-zero integers with no common factor. Thus,
b
2
=
a
{\displaystyle b{\sqrt {2}}=a}
. Squaring both sides yields 2b2 = a2. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is, a2 is even, which implies that a must also be even, as seen in the proposition above (in #Proof by contraposition). So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yields b2 = 2c2. But then, by the same argument as before, 2 divides b2, so b must be even. However, if a and b are both even, they have 2 as a common factor. This contradicts our previous statement that a and b have no common factor, so we must conclude that
2
{\displaystyle {\sqrt {2}}}
is an irrational number. To paraphrase: if one could write
2
{\displaystyle {\sqrt {2}}}
as a fraction, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.
=== Proof by construction ===
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example. It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property.
=== Proof by exhaustion ===