**Comparability of Sets:**

Two
sets A and B are said to be comparable if either of these happens:

**(i)**A Ì B

**(ii)**B Ì A

**(iii)**A = B

Similiarly
if neither of these above three exisits i.e., A Ë B, B Ë A and A≠B, then let A and
B are said to be incomparable.

Ex:
A={1,2,3} and B={1,2}

Hence
set A and B are comparable.

But
A ={1, 2, 3} and B={2, 3, 6, 7} are incomparable.

**Univerasal Set:**

Any
set which is super set of all the sets under consideration is known as the
universal set and is either denoted by Ω or S or U.

Ex:
Let A ={ 1, 2, 3} and B={3, 4,6, 9} and C={0, 1}

We
can take, S={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and Universal Set.

**Power Set:**

The
set or family of all the subsets of a given set A is said to be the power of A
and is expressed by P(A).

**Note:**

**(i)**Ø P(A) and A P(A) for all sets A.

**(ii)**The elements of P(A) are the subsets of A.

Ex: If, A = {1}, then P(A)+{Ø,{1}}

If A {1, 2}, then P(A)= {Ø, {1}, {2},
{1, 2}}.

Similarly if A={1, 2, 3}, then P(A)={Ø,
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

So trends, show that if A has n
elements then P(A) has 2

^{n}elements. Further to note that it is not only true for n=1, 2, 3, but it is true of all n.**Some Operations on Sets:**

**Union of Sets:**

The union of two sets A and B is set
of all those elements which are either in A or in B or in both. This set is
denoted by A U B and read as ‘A union B’.

Symbolically, A È B
={x : xA or x B}

Or a
È b
={x: x A U x B}

Example : A= {1,3,5,7,9} B= { 2,4,5,6,7} then A È B = { 1,2,3,4,5,6,7,9 }

**Intersection of Sets:**

The intersection of two sets A and B
is the set of all the elements, which are common in A and B. This set is
denoted by A (intersection) B and read as ‘A intersection of B’. i.e.,

**A Ç B={x: xA and x B}**

Or A Ç B ={x: xA Ç xB}

Example : A= {1,3,5,7,9} B= { 2,4,5,6,7} then A Ç B = { 5,7}

**Difference of Sets**

**:**

The difference of two sets A and B, is
the set of all those elements of A which are not elemtns of B. Sometimes, we
call differeence of sets as the relative components of B in A. It is denoted by
A (Intersection)B.

**i.**e., A-B ={x: xA and xB}

Similary B-A={x: xB and xA}

Example : A={ 1,3,5,6,8,10} B={ 2,3,4,5,6,7}, the A-B = {1,8,10}

**Symmetric Difference of Sets:**

If A and B are two sets then A ∆ B
=(A-B) È
(B-A) or (AÈB)
– (A Ç
B) is called Symmetric Difference of Sets.

**Complements of Set:**

The complement of a set A, also known
as absolute complement of A is the sets of all those element of the universal
sets which are not element of A. It is denoted by A

^{o}or A^{-1}.
Infact A

^{1}or A^{c}= U – A.
Or A

^{1}= {x: x U and xA}.**Properties:**

If A, B, C are three sets, then

**(i)**Idempotent laws: AÈ A=A, AÇ A=A.

**(ii)**Commutative laws: A È B=B È A, A Ç B = B Ç A.

**(iii)**Associative laws: (A È B)È C = AÈ(BÈC); (AÇ B)Ç C = A Ç (B Ç C).

**(iv)**Identity Laws: AUØ=A; AÇØ=Ø; AUµ=µ; AÇ µ=A.

**(v)**Complement laws: AUA

^{1}=µ; AÇA

^{1}=Ø; Ø

^{1}=µ,µ

^{1}=Ø.

**(vi)**Involution law: (A

^{1})

^{1}=A.

**(vii)**Demorgan laws: A-(BUC) = (A-B) inttersection (A-C);

A-(BÇC)=(A-B) U(A-C);

(AUB)

^{1}=A^{1}Ç B^{1}; (AÇ B)^{1}=A^{1}UB^{1}**(viii)**Distributive laws: AU(BÇ C) =(AUB)Ç (AUC);

A Ç (BUC) = (A Ç B) U (A Ç C).

**Some more important results:**

**i)**AÈB=Ø ó A=Ø and B=Ø.

**ii)**A-B=Ø ó A Í B

**iii)**A Ì A È B; B Ì A È B; A Ç B Ì A; A Ç B Ì B.

**iv)**A-B≠B-A; A-BÌA; A-B= AÇ B

^{1}; A-B = A ∆ (AÈB).

**v)**A Ì B ó B

^{1}Ì A

^{1}

**vi)**n(AÈB)= n(A)+n(B)-n(A intersection B).

**vii)**n(AÈBÈC)= n(A)+n(B)+n(C) – n(A Ç B) – n(B Ç C) – n(C Ç A) + n(A Ç B Ç C).

**VENN DIAGRAM AND SOME OF THE APPLICATIONS OF SET THEORY.**

Venn diagram is a pictorial
representation of sets.

A set is represented by circle or a closed
geometrial figure inside the universal set.

The Universal Set S, is represented by
a rectangular region. First of all we will represent the set or a statmenet
regarding sets with the help of the diagram or Venn Diagram. The shaded area
represents the set written.

**(a)**

**Subset:**

**(b)**

**Union of Sets:**Let A U B = B. Here, whole area represented by B represents A U B.

**A U B when A Ì B**

A U B when neither A ÌB nor BÌA

AÈB when A and B are disjoint events

**(c)**

**Intersection of Sets:**A (intersection) B represents the common area of A and B.

**AÇB when A Ì B , (AÇB=A)**

AÇB when neither A Ì B nor B Ì A

(AÇB=Ø) when A and B are disjoint sets.