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In mathematics, the '''commutative property''' states that a [[binary operation]] <math>*</math> on a set '''A''' is said to be commutative if for all <math>x,y</math> in '''A'''  we have <math>x*y=y*x</math>. An example of a commutative operation is [[addition]] in the set of [[real numbers]]. When a [[Group (mathematics)|group]]'s operation is commutative, it is said to be [[abelian]].  If <math>x*y=-(y*x)</math>, then the operation is said to be '''anticommutative'''.

In mathematics, the '''commutative property''' states that a binary operation <math>*</math> on a set '''A''' is said to be commutative if for all <math>x,y</math> in '''A'''  we have <math>x*y=y*x</math>. An Example of a commutative operation is addition in the [[real numbers]]. When a [[group (mathematics)|group]]'s operation is commutative, it is said to be [[abelian]].

In laymen's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2).  

In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the [[dot product]] is commutative, while the [[cross product]] is anticommutative.

It is as if the numbers are "commuting" from one place to another.

The commutative property implies the [[associative property]].

The commutative property does not imply the [[associative property]], nor vice versa.  For example, matrix multiplication is associative but not commutative.  Also, define the function <math>\Box</math> on integers by <math>a \Box b = 2(a + b)</math>.  Although, <math>\Box</math> is commutative, it is not associative, as <math>((1 \Box 0) \Box 0) = 2 \Box 0 = 4</math> but <math>(1 \Box (0 \Box 0)) = 1 \Box 0 = 2</math>.

[[Category:Mathematics]]

[[Category:Mathematics]]