# Commutative property

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.[1][2] A corresponding 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.[3]

Type An operation ${\displaystyle \circ }$ is commutative if and only if ${\displaystyle x\circ y=y\circ x}$ for each ${\displaystyle x}$ and ${\displaystyle y}$. This image illustrates this property with the concept of an operation as a "calculation machine". It doesn't matter for the output ${\displaystyle x\circ y}$ or ${\displaystyle y\circ x}$ respectively which order the arguments ${\displaystyle x}$ and ${\displaystyle y}$ have – the final outcome is the same. Law, Rule of replacement A binary operation is commutative if changing the order of the operands does not change the result. Definition 1: ${\displaystyle x*y=y*x\qquad \forall x,y\in S.}$ Propositional logic: ${\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}$ ${\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}$