# Binary operations

SET

Set is a collection of definite well defined objects.

Set is denoted by capital letter.

For example:

= {a, b, c, d, e}

BINARY OPERATIONS ON A SET
Let,

à a non-empty set

GXG = {(a ,b): a  G, b  G }.

Above line is read as: G cross G equal to (a, b) such that ‘a’ belongs to ‘G’, ‘b’ belongs to ‘G’.

If

f : GXG = G,

Above line is read as ‘f’ is such that G cross G is equal to G.

Here ‘f’ is an operation of ‘X’ on two groups ‘G’ and ‘G’.

The output of ‘GXG’ is also a ‘ G ‘ so this type of operation is known as Binary Operation on a set G.

And,

Operation ‘f’ on ‘G’ and ‘G’ can be denoted as ‘GfG’ , or ‘afb’ where (a  G, b   G).

+, x, etc symbols are used in Binary Operations.

Binary Operations examples:-

1)      a + b   G,  a, b  G.

Above line is read as ‘a’ plus ‘b’ belongs to ‘G’, for all ‘a’, ‘b’ belongs to ‘G’.

Here, ∀ à  for all.

2)      a * b = G,  a,b  G.

3)      Addition of natural numbers is also a natural number.

Natural number are also known as all non-negative or positive numbers (0,1,2,3,4……).

If, N à Set of natural numbers

A + b  N,  a, b  N.

Above line is read as ‘a’ plus ‘b’ belongs to ‘N’, for all ‘a’, ‘b’ ‘N’.

4)      Subtraction is not binary operation on N (natural numbers).

Nà Set of natural numbers.

3 – 5 = – 2  N, whereas 3, 5  N.

Above line is read as three minus five is not belongs to ‘N’, whereas three, five belongs to ‘N’.

5)      Subtraction is binary operation on I (integer numbers).

I –> Set of integer numbers

3 – 5 = -2  I,   a, b  I.