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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:**

- The operation of addition is a binary operation on the set of natural numbers.
- 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.
- The operation of multiplication is a binary operation on the set of natural numbers, set of integers and set of complex numbers.
- 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= {a_{1},a_{2},a_{3},….a_{n}}. A binary operation * on A can be described by means of table as shown in fig:

* | a_{1} | a_{2} | a_{3} | a_{n} | |

a_{1} | a_{1}*a_{1} | ||||

a_{2} | a_{2}*a_{2} | ||||

a_{3} | a_{3}*a_{3} | ||||

a_{n} | a_{n}*a_{n} |

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 |