inverse element

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Suppose that, for the binary operation ○ on the set S, there is a neutral element e. An element a′ is an inverse (or inverse element) of the element a if aa′=a′○ a=e. If the operation is called multiplication, the neutral element is normally called the identity element and may be denoted by 1. Then the inverse a′ may be called a multiplicative inverse of a and be denoted by a−1, so that aa−1=a−1a=1 (or e). If the operation is addition, the neutral element is denoted by 0, and the inverse a′ may be called an additive inverse (or a negative) of a and be denoted by −a, so that a+(−a)=(−a)+a=0. See also group.

Subjects: Mathematics.

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