Commutativity

A map or binary operation from a set to a set is said to be commutative if,<br>

(A common example in school-math is the '+' "function": , thus the '+' "function" is commutative)

Otherwise, the operation is noncommutative. Additionally, if a particular pair of elements satisfies the equation above then it is said that the two elements commute.

Properties

Every element belonging to the domain of a binary operation commutes with itself and,
If is at least a group, each one of its elements commutes with the identity, with its own inverse, and with its powers.
Commutativity can be another name for symmetry. For example, suppose we solve a problem involving parameters and , and determine the solution in the form . If there exists a subset of values for and where the two values can be exchanged without affecting the value of 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).

Center of noncommutative operations

In algebra, the subset of the domain of a binary operation on which an operation is commutative is sometimes called center .

and

Subtraction of real numbers is noncommutative, since, but, And so all pairs constitutes the center of the subtraction operation.

Examples

Commutative operations

The most well-known examples of commutative binary operations are

addition of real numbers is commutative since For example 4 + 5 = 5 + 4, since both expressions evaluate to 9.
multiplication of real numbers is commutative since For example 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.

Noncommutative operations

Among the noncommutative binary operations are

subtraction ,
division ,
exponentiation ,
function composition ,
tetration ,
matrix multiplication, since for example
quaternion multiplication, since

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.

Mathematical structures and commutativity

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.

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