binary operation

Summary

Properties

Common binary operations

division is not a binary operation in since division by 0 is undefined

Multiplication set

• makes division a binary operation

but addition and subtraction are no longer binary operations, since

Concept

Binary operation

• binary function that maps any two elements in a set to another element in the same set
• implies a closure of the set under the operator

Multiplication table

• only possible for finite sets
• symmetry along the diagonal -> abelian

Identities

• neutral element
• operated alongside another element, the element is returned

Application

Multiplication table

Associative but not abelian

Abelian but not associative

Identity