Operations on Sets:-

Just as operations of addition, subtraction etc., are performed on numbers, the
operations of unions, intersection etc., are performed on sets. We are already familiar with
them. A review of the main rules is given below: –
Union of two sets: The Union of two sets $A$ and $B$ , denoted by $A\cup B$ , is the set of all elements,
which belong to $A$ or $B$ .

Symbolically;

$A\cup B$ = ${{\;x/x\in A\;\vee\;x\in B}}$
Thus if $A$ = $\left { 1,2,3 \right }$ , $B=\left ] 2,3,4,5 \right }$ [/latex]

$A\cup B$ = $\left { 1,2,3,4,5 \right }$

Intersection of two sets:

The intersection of two sets A and B, denoted by $A\cap B$ , is the set
of all elements, which belong to both $A$ and $B$ .

Symbolically;

Disjoint Sets:

If the intersection of two sets is the empty set then the sets are said to be
disjoint sets.

For example;

If $S_{1}$ = The set of odd natural numbers and $S_{2}$ = The set of even natural numbers, then $S_{1}$ and $S_{2}$
are disjoint sets.

The set of arts students and the set of science students of a
college are disjoint sets.

Overlapping sets:

If the intersection of two sets is non-empty but neither is a subset of the
other, the sets are called overlapping sets, e.g., if
L = {2,3,4,5,6} and M= {5,6,7,8,9,10}, then L and M are two overlapping sets

Complement of a set:

The complement of a set $A$ , denoted by $A^{'}$ or $A^{c}$
relative to the universal
set $U$ is the set of all elements of $U$ , which do not belong to $A$ .
Symbolically:

$A^{'}$ =\left { x/x\in U\wedge x\notin A \right }[/latex]
For example, if $U=N$ , then $E^{'} = O$ and $O^{'}=E$

Example 1:

If $U$ = set of alphabets of English language,
$C$ = set of consonants,
$W$ = set of vowels, then $C^{'}= W$ and $W^{'}= C$

Difference of two Sets:

The Difference set of two sets $A$ and $B$ denoted by $A-B$ consists of
all the elements which belong to $A$ but do not belong to $B$ .
The Difference set of two sets $B$ and $A$ denoted by $B-A$ consists of all the elements, which
belong to $B$ but do not belong to $A$ .
Symbolically, A-B = { } xx A x B ∈ ∧∉ and B-A = { } xx B x A ∈ ∧∉
Example 2: If A = {1,2,3,4,5}, B = {4,5,6,7,8,9,10}, then
A-B = {1,2,3} and B-A = {6,7,8,9,10}

Intersection of two sets:

Disjoint Sets:

Overlapping sets:

Complement of a set:

Difference of two Sets:

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