Binary operation: Difference between revisions
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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 | 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> |
Revision as of 12:53, 12 December 2008
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: