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 .


A binary operation may satisfy further conditions.

  • Commutative:
  • Associative:
  • Alternative:
  • Power-associative:

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

  • Neutral element I: for all x
  • Absorbing element O: for all x