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Concept
Commutative property
Summary
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.
A binary operation on a set S is called commutative if
An operation that does not satisfy the above property is called non-commutative.
One says that x commutes with y or that x and y commute under if
In other words, an operation is commutative if every two elements commute.
Addition and multiplication are commutative in most number systems, and, in particular, between natural numbers, integers, rational numbers, real numbers and complex numbers. This is also true in every field.
Addition is commutative in every vector space and in every algebra.
Union and intersection are commutative operations on sets.
"And" and "or" are commutative logical operations.
Some noncommutative binary operations:
Equation xy = yx
Division is noncommutative, since .
Subtraction is noncommutative, since . However it is classified more precisely as anti-commutative, since .
Exponentiation is noncommutative, since .
Official source
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