Editing Associative property (section)

==Examples==
[[File:Associativity of real number addition.svg|thumb|The addition of real numbers is associative.]]
Some examples of associative operations include the following.

{{unordered list
|1= The [[string concatenation|concatenation]] of the three strings <code>"hello"</code>, <code>" "</code>, <code>"world"</code> can be computed by concatenating the first two strings (giving <code>"hello "</code>) and appending the third string (<code>"world"</code>), or by joining the second and third string (giving <code>" world"</code>) and concatenating the first string (<code>"hello"</code>) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).
|2= In [[arithmetic]], [[addition]] and [[multiplication]] of [[real number]]s are associative; i.e.,

<math display="block">
\left.
\begin{matrix}
(x+y)+z=x+(y+z)=x+y+z\quad
\\
(x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \,
\end{matrix}
\right\}
\mbox{for all }x,y,z\in\mathbb{R}.
</math>

Because of associativity, the grouping parentheses can be omitted without ambiguity.
|3= The trivial operation {{math|1={{var|x}} ∗ {{var|y}} = {{var|x}}}} (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation <math>x \circ y = y</math> (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.
|4= Addition and multiplication of [[complex number]]s and [[quaternion]]s are associative. Addition of [[octonion]]s is also associative, but multiplication of octonions is non-associative.
|5= The [[greatest common divisor]] and [[least common multiple]] functions act associatively.

<math display="block">
\left.
\begin{matrix}
\operatorname{gcd}(\operatorname{gcd}(x,y),z)=
\operatorname{gcd}(x,\operatorname{gcd}(y,z))=
\operatorname{gcd}(x,y,z)\ \quad
\\
\operatorname{lcm}(\operatorname{lcm}(x,y),z)=
\operatorname{lcm}(x,\operatorname{lcm}(y,z))=
\operatorname{lcm}(x,y,z)\quad
\end{matrix}
\right\}\mbox{ for all }x,y,z\in\mathbb{Z}.
</math>
|6= Taking the [[intersection (set theory)|intersection]] or the [[union (set theory)|union]] of [[Set (mathematics)|sets]]:
<math display="block">
\left.
\begin{matrix}
(A\cap B)\cap C=A\cap(B\cap C)=A\cap B\cap C\quad
\\
(A\cup B)\cup C=A\cup(B\cup C)=A\cup B\cup C\quad
\end{matrix}
\right\}\mbox{for all sets }A,B,C.
</math>
|7= If {{mvar|M}} is some set and {{mvar|S}} denotes the set of all functions from {{mvar|M}} to {{mvar|M}}, then the operation of [[function composition]] on {{mvar|S}} is associative:<math display="block">(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h\qquad\mbox{for all }f,g,h\in S.</math>
|8= Slightly more generally, given four sets {{mvar|M}}, {{mvar|N}}, {{mvar|P}} and {{mvar|Q}}, with {{math|{{var|h}} : {{var|M}} → {{var|N}}}}, {{math|{{var|g}} : {{var|N}} → {{var|P}}}}, and {{math|{{var|f}} : {{var|P}} → {{var|Q}}}}, then

<math display="block">(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h</math>

as before. In short, composition of maps is always associative.
|9= In [[category theory]], composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms.
|10= Consider a set with three elements, {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}. The following operation:

{{wikitable| class="wikitable" style="text-align:center"
|-
! × !! {{mvar|A}} !! {{mvar|B}} !! {{mvar|C}}
|-
! {{mvar|A}}
| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}
|-
! {{mvar|B}}
| {{mvar|A}} || {{mvar|B}} || {{mvar|C}}
|-
! {{mvar|C}}
| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}
}}

is associative. Thus, for example, {{math|1={{var|A}}({{var|B}}{{var|C}}) = ({{var|A}}{{var|B}}){{var|C}} = {{var|A}}}}. This operation is not commutative.
|11= Because [[Matrix (mathematics)|matrices]] represent [[linear map|linear function]]s, and [[matrix multiplication]] represents function composition, one can immediately conclude that matrix multiplication is associative.<ref>{{cite web|url=http://www.khanacademy.org/math/linear-algebra/matrix-transformations/composition-of-transformations/v/matrix-product-associativity|title=Matrix product associativity|publisher=Khan Academy|access-date=5 June 2016}}</ref>
|12= For [[real number]]s (and for any [[totally ordered set]]), the minimum and maximum operation is associative: <math display="block">\max(a, \max(b, c)) = \max(\max(a, b), c) \quad \text{ and } \quad \min(a, \min(b, c)) = \min(\min(a, b), c).</math>

}}