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 '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 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. The commutative property (or commutative law) is a property generally associated with binary operations and functions. If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation. The term 'commutative' is used in several related senses. Two well-known examples of commutative binary operations: Some noncommutative binary operations: Division is noncommutative, since 1 ÷ 2 ≠ 2 ÷ 1 {displaystyle 1div 2 eq 2div 1} . Subtraction is noncommutative, since 0 − 1 ≠ 1 − 0 {displaystyle 0-1 eq 1-0} . However it is classified more precisely as anti-commutative, since 0 − 1 = − ( 1 − 0 ) {displaystyle 0-1=-(1-0)} . Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for (A ⇒ B) = (¬A ∨ B) and (B ⇒ A) = (A ∨ ¬B) are