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1.10: Binary Operation

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    A binary operation on a set S is a mapping f from the Cartesian product S × S to S. A mapping from K x S to S, where K need not be S, is called an external binary operation.

    Many binary operations are commutative (i.e. f(a,b) = f(b,a) holds for all a, b in S) or associative (i.e. f(f(a,b), c) = f(a, f(b,c)) holds for all a,b,c in S). Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations.

    Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@), and composition.

    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). Sometimes they are even written just by concatenation: ab.

    This page titled 1.10: Binary Operation was last modified on Sat, 01 Jul 2023 00:30:19 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography via source content that was edited to the style and standards of the LibreTexts platform.