kb/data/en.wikipedia.org/wiki/Abuse_of_notation-1.md

title	chunk	source	category	tags	date_saved	instance
Abuse of notation	2/2	https://en.wikipedia.org/wiki/Abuse_of_notation	reference	science, encyclopedia	2026-05-05T07:22:55.629926+00:00	kb-cron

    (
    E
    ×
    F
    )
    ×
    G
    =
    E
    ×
    (
    F
    ×
    G
    )
    =
    E
    ×
    F
    ×
    G
  

{\displaystyle (E\times F)\times G=E\times (F\times G)=E\times F\times G}

. But this is strictly speaking not true: if

    x
    ∈
    E
  

{\displaystyle x\in E}

,

    y
    ∈
    F
  

{\displaystyle y\in F}

and

    z
    ∈
    G
  

{\displaystyle z\in G}

, the identity

    (
    (
    x
    ,
    y
    )
    ,
    z
    )
    =
    (
    x
    ,
    (
    y
    ,
    z
    )
    )
  

{\displaystyle ((x,y),z)=(x,(y,z))}

would imply that

    (
    x
    ,
    y
    )
    =
    x
  

{\displaystyle (x,y)=x}

and

    z
    =
    (
    y
    ,
    z
    )
  

{\displaystyle z=(y,z)}

, and so

    (
    (
    x
    ,
    y
    )
    ,
    z
    )
    =
    (
    x
    ,
    y
    ,
    z
    )
  

{\displaystyle ((x,y),z)=(x,y,z)}

would mean nothing. However, these equalities can be legitimized and made rigorous in category theory—using the idea of a natural isomorphism. Another example of similar abuses occurs in statements such as "there are two non-Abelian groups of order 8", which more strictly stated means "there are two isomorphism classes of non-Abelian groups of order 8".

=== Equivalence classes === Referring to an equivalence class of an equivalence relation by

    x
  

{\displaystyle x}

instead of

    [
    x
    ]
  

{\displaystyle [x]}

is an abuse of notation. Formally, if a set

    X
  

{\displaystyle X}

is partitioned by an equivalence relation

    ∼
  

{\displaystyle \sim }

, then for each

    x
    ∈
    X
  

{\displaystyle x\in X}

, the equivalence class

    {
    y
    ∈
    X
    
      |
    
    y
    ∼
    x
    }
  

{\displaystyle \{y\in X|y\sim x\}}

is denoted

    [
    x
    ]
  

{\displaystyle [x]}

. But in practice, if the remainder of the discussion is focused on the equivalence classes rather than the individual elements of the underlying set, then it is common to drop the square brackets in the discussion. For example, in modular arithmetic, a finite group of order

    n
  

{\displaystyle n}

can be formed by partitioning the integers via the equivalence relation "

    x
    ∼
    y
  

{\displaystyle x\sim y}

if and only if

    x
    ≡
    y
     
    (
    
      m
      o
      d
    
     
    n
    )
  

{\displaystyle x\equiv y\ (\mathrm {mod} \ n)}

". The elements of that group would then be

    [
    0
    ]
    ,
    [
    1
    ]
    ,
    …
    ,
    [
    n
    −
    1
    ]
  

{\displaystyle [0],[1],\dots ,[n-1]}

, but in practice they are usually denoted simply as

    0
    ,
    1
    ,
    .
    .
    .
    ,
    n
    −
    1
  

{\displaystyle 0,1,...,n-1}

. Another example is the space of (classes of) measurable functions over a measure space, or classes of Lebesgue integrable functions, where the equivalence relation is equality "almost everywhere".

== Subjectivity == The terms "abuse of language" and "abuse of notation" depend on context. Writing "f : A → B" for a partial function from A to B is almost always an abuse of notation, but not in a category theoretic context, where f can be seen as a morphism in the category of sets and partial functions.

== See also == Mathematical notation Misnomer

== References ==