Binary operation

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A binary operation on a set S is a function of the form f : S × S -> S.

Binary operations are often written using infix notation such as a * b, a + b or a.b rather than by functional notation of the form f(a,b). Many binary operations of interest are commutative or associative.

Typical examples of binary operations are the addition and multiplication of numbers as well as composition of functions: start with a set M and consider two functions r : M -> M and s : M -> M. Then ros : M -> M is the function defined by (ros)(x) = r(s(x)) for all x in M. This defines a binary operation o on the set S containing all functions from M to M.

See also binary function.