Commutative property

The commutative (Latin commutare "Exchange " ), in German Vertauschungsgesetz, is a rule of mathematics. If it is true, the arguments of an operation can be reversed without changing anything in the result. Mathematical operations that are subject to commutative is called commutative.

The commutative forms with the associative law and distributive basic rules of algebra.

Formal definition

Let and quantities. A binary operation is called commutative if the equality holds for all.

Examples and counter-examples

Real Numbers

For real numbers always

The operations of addition and multiplication are commutative so. The first formula is also called the commutative property of addition, the second commutative property of multiplication. The subtraction and division of real numbers, however, are not commutative operations. The exponentiation is not commutative (example: )

The oldest known form of the commutative law of addition is the Sumerian tale of the wise wolf and the nine stupid wolves.

Scalar products

• The scalar product in a real vector space is commutative, so it is always valid.
• The scalar product in a complex vector space, however, is not commutative, it is rather, where the overline denotes complex conjugation.

Set operation

In set theory, the union and the intersection are commutative operations; for amounts so always applies

However the difference is not commutative, in non-trivial cases (i.e. when and ) are therefore and various amounts.

Matrix

The addition of matrices over a ring or body is commutative. The matrix multiplication is not commutative, however, in general. For the product of a square matrix A with its inverse matrix (gives the identity matrix ) is the commutativity of multiplication, however, given, as for the multiplication of any ( square ) matrix with the identity matrix. Also commutative is the scalar one ( arbitrary) matrix by a scalar, and the multiplication in the lower ring of the diagonal matrices.

A group of matrices which commute with respect to multiplication, is called Abelian.

Propositional logic

In propositional logic applies to the connectives:

• ( "Or" ) is commutative
• ( "And") is commutative
• ( " Logical equivalence " ) is commutative.
• ("If ..., then ..." (see implication ) is not commutative.

Other examples

Other examples of non-commutative operations are the cross product in vector spaces, or the multiplication of quaternions.

Commutativity is also an important fundamental property in group theory and quantum mechanics.

Antikommutativität

In some structures with two operations, for example in the cross product in vector spaces, does not apply the commutative, but instead a kind of opposite:

Comments

The commutativity that allows swapping of arguments in an operation has similarities to the symmetry property of relations, which put simply swapping the elements compared with respect to the relation allows, for example, follows from x R y implies y R x.

An alternative possibility of " To - stapling " offers the flexibility of law for a link *