Binary operations

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set. More formally, a binary operation is an operation of arity two whose two domains and one codomain are the same set or subsets thereof. Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.

More precisely, a binary operation on a set S is a map which sends elements of the Cartesian product S × S to S:

Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure). If f is not a function, but is instead a partial function, it is called a partial binary operation. For instance, division of real numbers is a partial binary operation, because one can't divide by zero: a/0 is not defined for any real a. Note however that both in algebra and model theory the binary operations considered are defined on all of S × S.

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