From Citizendium - Reading time: 2 min
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 ![{\displaystyle (x,y)\mapsto x\star y,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca813077bb83a34d5e606e09ac59296e98d0743)
using operator notation rather than functional notation, which would call for writing
.
Properties[edit]
A binary operation may satisfy further conditions.
- Commutative:
![{\displaystyle x\star y=y\star x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9fba4eb208c9d5aa4d84de95aa7def7414c726)
- Associative:
![{\displaystyle (x\star y)\star z=x\star (y\star z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b211723abebcb4a91912ae1fd2c3457a3980a8fa)
- Alternative:
![{\displaystyle (x\star y)\star y=x\star (y\star y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11a97f81a7b6809b0c7f818a1dde87f7f64a11e9)
- Power-associative:
![{\displaystyle (x\star x)\star x=x\star (x\star x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9073fe848283ae971cd0c03db884f31b626126d7)
Special elements which may be associated with a binary operation include: