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.