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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,....an}. 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 aj*ak.

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





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