Editing Commutative property (section)

== Examples ==
[[File:Commutative Addition.svg|thumb|The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.]]

=== Commutative operations ===
[[File:Vector Addition.svg|thumb|The addition of vectors is commutative, because <math>\vec a+\vec b=\vec b+ \vec a.</math>]]
* [[Addition]] and [[multiplication]] are commutative in most [[number system]]s, and, in particular, between [[natural number]]s, [[integer]]s, [[rational number]]s, [[real number]]s and [[complex number]]s. This is also true in every [[field (mathematics)|field]].{{sfn|Rosen|2013|loc = See the [https://books.google.com/books?id=-oVvEAAAQBAJ&pg=SL1-PA1 Appendix I]}}
* Addition is commutative in every [[vector space]] and in every [[algebra over a field|algebra]].{{sfn|Sterling|2009|p=[https://books.google.com/books?id=PsNJ1alC-bsC&pg=PA248 248]}}
* [[Union (set theory)|Union]] and [[intersection (set theory)|intersection]] are commutative operations on [[set (mathematics)|set]]s.{{sfn|Johnson|2003|p=[http://books.google.com/books?id=-pZX2KS2KqMC&pg=PA642 642]}}
* "[[And (logic)|And]]" and "[[or (logic)|or]]" are commutative [[logical operation]]s.{{sfn|O'Regan|2008|p=[https://books.google.com/books?id=081H96F1enMC&pg=PA33 33]}}

=== Noncommutative operations ===
* [[Division (mathematics)|Division]] is noncommutative, since <math>1 \div 2 \neq 2 \div 1</math>. [[Subtraction]] is noncommutative, since <math>0 - 1 \neq 1 - 0</math>. However it is classified more precisely as [[Anticommutativity|anti-commutative]], since <math>x - y = - (y - x)</math> for every {{tmath|x}} and {{tmath|y}}. [[Exponentiation]] is noncommutative, since <math>2^3\neq3^2</math> (see [[Equation xy = yx{{!}}Equation ''x<sup>y</sup>'' = ''y<sup>x</sup>'']].{{sfn|Posamentier|Farber|Germain-Williams|Paris|2013|p=[https://books.google.com/books?id=VfCgAQAAQBAJ&pg=PA71 71]}}
* Some [[truth function]]s are noncommutative, since their [[truth table]]s are different when one changes the order of the operands.{{sfn|Medina|Ojeda-Aciego|Valverde|Vojtáš|2004|p=[https://books.google.com/books?id=yAH6BwAAQBAJ&pg=PA617 617]}} For example, the truth tables for {{math|(A ⇒ B) {{=}} (¬A ∨ B)}}  and {{math|(B ⇒ A) {{=}} (A ∨ ¬B)}} are
: {{aligned table
 | class=wikitable
 | style=text-align:center; width:20%;
 | cols=4
 | col3style=width:30%;
 | col4style=width:30%;
 | row1header=yes
 | {{math|A}} | {{math|B}} | {{math|A ⇒ B}} | {{math|B ⇒ A}} 
 | F | F | T | T
 | F | T | T | F
 | T | F | F | T
 | T | T | T | T
 }}
* [[Function composition]] is generally noncommutative.{{sfn|Tarasov|2008|p=[http://books.google.com/books?id=pHK11tfdE3QC&pg=PA56 56]}} For example, if <math>f(x)=2x+1</math> and <math>g(x)=3x+7</math>. Then <math display="block">(f \circ g)(x) = f(g(x)) = 2(3x+7)+1 = 6x+15</math> and <math display="block">(g \circ f)(x) = g(f(x)) = 3(2x+1)+7 = 6x+10.</math>
* [[Matrix multiplication]] of [[square matrices]] of a given dimension is a noncommutative operation, except for {{tmath|1\times 1}} matrices. For example:{{sfn|Cooke|2014|p=[https://books.google.com/books?id=b_iJAwAAQBAJ&pg=PA7 7]}} <math display="block">
  \begin{bmatrix}
    0 & 2 \\
    0 & 1
  \end{bmatrix} =
  \begin{bmatrix}
    1 & 1 \\
    0 & 1
  \end{bmatrix} 
  \begin{bmatrix}
    0 & 1 \\
    0 & 1
  \end{bmatrix} \neq
  \begin{bmatrix}
    0 & 1 \\
    0 & 1
  \end{bmatrix} 
  \begin{bmatrix}
    1 & 1 \\
    0 & 1
  \end{bmatrix} =
  \begin{bmatrix}
    0 & 1 \\
    0 & 1
  \end{bmatrix}
</math>
* The vector product (or [[cross product]]) of two vectors in three dimensions is [[anticommutativity|anti-commutative]]; i.e., <math> \mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b}) </math>.{{sfn|Haghighi|Kumar|Mishev|2024|p=[http://books.google.com/books?id=D2b8EAAAQBAJ&pg=PA118 118]}}