Aptitude for Placement Papers

Aptitude for Placement Papers - Set Theory -2

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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 2ⁿ 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 : x

A or x

Or a È b ={x: x

A U x

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: x

A and x

Or A Ç B ={x: x

A Ç x

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: x

A and x

Similary B-A={x: x

B and x

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 A¹ or A^c = U – A.

Or A¹= {x: x

U and x

A}.

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: AUA¹=µ; AÇA¹=Ø; Ø¹=µ,µ¹=Ø.

(vi) Involution law: (A¹)¹=A.

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

A-(BÇC)=(A-B) U(A-C);

(AUB)¹=A¹Ç B¹; (AÇ B)¹=A¹UB¹

(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 B¹; A-B = A ∆ (AÈB).

v) A Ì B ó B¹Ì A¹

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.