Difference between revisions of "Binary operation"

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imported>David E. Volk
m (subpages)
imported>Richard Pinch
(→‎Properties: Added idempotent)
 
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* [[Power-associative]]: <math>(x \star x) \star x = x \star (x \star x)</math>
* [[Power-associative]]: <math>(x \star x) \star x = x \star (x \star x)</math>


Special elements which may be associated with a binary operations include:
Special elements which may be associated with a binary operation include:
* [[Neutral element]] ''I'': <math>I \star x = x \star I = x</math> for all ''x''
* [[Neutral element]] ''I'': <math>I \star x = x \star I = x</math> for all ''x''
* [[Absorbing element]] ''O'': <math>O \star x = x \star O = O</math> for all ''x''
* [[Absorbing element]] ''O'': <math>O \star x = x \star O = O</math> for all ''x''
* [[Idempotent element]] ''E'': <math>E \star E = E</math>

Latest revision as of 12:53, 12 December 2008

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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.

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

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