Let S be a non-void set. A function from S×S to S is called a binary operation on S.
f:S×S S is binary operation on set S
A binary function f on a set S×S associates each order pair (a,b) of elements of S×S to a unique element f(a,b) of S.
Instead of writing f(a,b) for the image of an ordered pair we can write a f b.
a f b = f(a,b)
Binary operations are represented by the symbols *, etc. instead of letter f,g etc.
Instead S×S only Set S is mentioned in binary operation but it is to be understood as S×S.
Thus a binary operation * on a set S associates each ordered pair (a,b) of elements of S to a unique element a*b of S. (a*b is an element in S that is obtained by the relation a*b -- * represents relation on set S and set S)
Example: Addition of two natural numbers.
(a,b) is an element of N×N (a and b are natural numbers)
a+b is also a natural number and is also an element of N.
Addition of natural numbers is f:N×NN a binary operation.
Types of binary operations
Commutative binary operation.
Associative binary operation.
Distributive binary operation
Identity and inverse elements
Inverse of an element
A binary operation on finite set can be completely described by means of a table known as a composition table.