# Binary operation

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In mathematics, a **binary operation** on a set is a function of two variables which assigns a value to any pair of elements of the set: principal motivating examples include the arithmetic and elementary algebraic operations of addition, subtraction, multiplication and division.

Formally, a binary operation on a set *S* is a function on the Cartesian product

- given by

using operator notation rather than functional notation, which would call for writing .

## Properties

A binary operation may satisfy further conditions.

Special elements which may be associated with a binary operation include:

- Neutral element
*I*: for all*x* - Absorbing element
*O*: for all*x* - Idempotent element
*E*: