**SET**

Set is a collection of definite well defined objects.

Set is denoted by capital letter.

For example:

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

**BINARY OPERATIONS ON A SET**

Let,

G 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.