Binary Operation

Consider a non-empty set A and α function f: AxA→A is called a binary operation on A. If * is a binary operation on A, then it may be written as a*b.

A binary operation can be denoted by any of the symbols +,-,*,⨁,△,⊡,∨,∧ etc.The value of the binary operation is denoted by placing the operator between the two operands.

Example:

1. The operation of addition is a binary operation on the set of natural numbers.
2. The operation of subtraction is a binary operation on the set of integers. But, the operation of subtraction is not a binary operation on the set of natural numbers because the subtraction of two natural numbers may or may not be a natural number.
3. The operation of multiplication is a binary operation on the set of natural numbers, set of integers and set of complex numbers.
4. The operation of the set union is a binary operation on the set of subsets of a Universal set. Similarly, the operation of set intersection is a binary operation on the set of subsets of a universal set.

N-ARY Operation:

A function f: AxAx………….A→A is called an n-ary operation.

Tables of Operation:

Consider a non-empty finite set A= {a₁,a₂,a₃,….a_n}. A binary operation * on A can be described by means of table as shown in fig:

*	a1	a2	a3		an
a1	a1*a1
a2		a2*a2
a3			a3*a3
an					an*an

The empty in the jth row and the kth column represent the elements a_j*a_k.

Example: Consider the set A = {1, 2, 3} and a binary operation * on the set A defined by a * b = 2a+2b.

Represent operation * as a table on A.

Solution: The table of the operation is shown in fig:

*	1	2	3
1	4	6	8
2	6	8	10
3	8	10	12