Commutativity

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For other meanings of commutation, see commutation (disambiguation).

A map or binary operation $f\colon A \times A \rightarrow B$ is said to be commutative when, for any y and any z in A
$f (y, z) = f (z, y)$ ; otherwise, the operation is noncommutative.

For example, multiplication of real numbers is commutative since

y z = z y

for all real numbers y and z. On the other hand, subtraction of real numbers is noncommutative, since

y - z = z - y

if and only if y and z are identical.

Additionally, if

f (y, z) = f (z, y)

for a particular pair of elements y and z, then y and z are said to commute. Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers.

The most well-known examples of commutative binary operations are addition and multiplication of real numbers; for example:

4 + 5 = 5 + 4 (since both expressions evaluate to 9)
2 × 3 = 3 × 2 (since both expressions evaluate to 6)

Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. In each case, these operations are commutative over their entire domains.

Among the noncommutative binary operations are subtraction (a − b), division (a/b), exponentiation (a^b), function composition (f o g), tetration (a↑↑b), matrix multiplication, and quaternion multiplication.

A real life example of noncommutativity is the Rubik's Cube: for example, twisting the front face clockwise, the top face clockwise and the front face counterclockwise (FUF') does not yield the same result as twisting the front face clockwise, then counterclockwise and finally twisting the top clockwise (FF'U). The twists don't commute. This is studied in group theory.

The subset of the domain on which an operation is commutative is sometimes called the center in algebra.

An abelian group is a group whose group operation is commutative. A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) In a field both addition and multiplication are commutative.

Commutativity can be another name for symmetry. That is, suppose we solve a problem involving parameters x and y, and determine that the solution is equal to $f (x, y)$ . If there exists a subset of values for x and y where the two values can be exchanged without affecting the function, the problem is symmetric. Many symmetries arise naturally in mathematics out of simpler symmetries, and are commonly found useful for particular kinds of proofs (see WLOG).