Set Theory

CPT Section D Quantitative Aptitude Chapter 7

Brijeshwar Prasad Gupta

Learning Objectives

Number system

Set Theory

Set operations

Product of Sets

MCQ

Number system

Natural numbers:- N

• N = {1,2,3……..}

• 0 N, -3 N, ⅔ N,√2 N

Whole numbers:- W

• W = {0,1,2,3…….}

∉ ∉ ∉ ∉

Number system

Integers:- I or Z • I = {----- -3,-2,-1,0,1,2,3……..}

• Positive integers:- I+ • I+ = {1,2,3………}

• Negative integers:- I- • I- = {-1,-2,-3………}

Remark:- 0 ∉ I+, 0 ∉ I- but 0 ∈ I

Number system

Prime numbers :- P • P = {x:x is divisible by either one or

itself not by any other number except 1} • or

• P = {2,3,5,7,11,13,17……….}

Number system

Rational numbers:- Q • Q = {x:x can be expressed in the form

of P/q, where p & q I but q ≠ 0, p & q are prime to each other}

Irrational numbers:- Q’ • Q’ = {x:x can not be expressed in the

form of P/q}

∉

Number system

Real numbers :- R • R = A set of all rational and all irrational numbers are real numbers

• i.e. • R = Q U Q’

Set Theory

A Collection of Well defined objects • A set of vowels of English alphabet • A set of even numbers less than

100 • A set of multiple of 5

Set Theory

A set of vowels of English alphabet

• A = {a,e,i,o,u}

Set Theory

A set of even numbers less than 100

• B = {2,4,6……98}

Set Theory

A set of multiple of 5 • C = {5,10,15……..}

Set Theory

Representation of sets :- A,B,C………

And members are placed only in { }

Methods of describing a set • Tabular (Roster, Enumeration ) Method

• Selector ( Builder , Rule ) Method • Venn Diagram

Set Theory

• A = { a, e, i, o, u} • B = {2,4,6,8……98} • C = {5,10,15………}

Tabular method

Set Theory

Set Theory

Venn Diagram :- Diagrammatical representation by closed polygon usually by Circle & Rectangle

Types of sets

Finite & infinite set

Singleton set

Null or void set

Equal set

Equivalent set

Types of Sets

Joint & disjoint set

Sub set

Family of sets

Power set

Universal set

Cardinal number

Finite set

A set whose elements are countable • A = {p, q, r, s} • B = {1,3,5,7…..1000} • C = {x:x = 5n where n N} ∉

Infinite set

A set whose elements are uncountable. • A = {2,4,6,8……..} • B = {x:x is n odd number} • C = {x:x = 2n where n R} ∉

Singleton set

A set in which there is only single element.

• A = {p} • B = {x:x is a perfect square where 20<x <30} • C = {x:x is neither positive nor negative} • D= {x:x is an even prime number }

Null or void set

Equal set

Two sets are said to be equal if they have same elements • A = {a, e, i, o, u} • B = {a, i, u, o, e} • C = {a, e, e, e, i, i, o, u} • A = B = C • Contd….

Equal set:Continued

P = {x:x is a letter of word “march”}

Q = {x:x is a letter of word “charm”}

P = Q

Remark:- Repetation and arrangement of element does not effect equality of sets.

Equivalent set

Two sets are said to be equivalent if they have same number of elements

• A = {a, e, i, o, u} • B = {1,3,5,7,9,} • A Ξ B

Joint set

If two sets have some common elements than they are joint sets

• A = {a, e, i, o, u} • B = { a, b, c, d, e, f} • i.e. A ∩ B ≠ ϕ

Disjoint Set

Two sets are disjoint if they have no common element

• A = { a, e, i, o, u} • B = { p, q, r, s} • i.e. A ∩ B = ф

Cardinal number

Representation of number of elements in a given set. It is represented by n (A) • A = {a, e, i, o, u} • n (A) = 5

Sub Set

Sub Set: Remarks

Sub set: Remarks

6. All possible sub sets of a given set contains “n” elements are 2n.

Number of elements Number of sub sets 1. 21 = 2 2. 22 = 4 3. 23 = 8 etc.

Sub Set

A = {a, b, c}

Total subsets are 8

{a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}, ϕ

Family of Sets

A set of sets is family of set • A = {{a, b}, {2,4,6}, {p, q, r, s}}

Power set

A family of set contains all possible subsets of a given set • A = {1,3,5} • P(A) =

{{1},{3},{5},{1,3},{3,5},{5,1},{1,3,5} ,ϕ}

Universal set

A set contains all the elements of concerning sets. It is represented by either U or E • A = {2,4,6,8} • B = {1,3,5,7,9} • C = {5,10,15,20} • E = {1,2,3,……..20}

Set Operations

Union operation

Intersection operation

Compliment operation

Difference of sets

Symmetric difference

Product of sets

Union Operation

Union of two sets is represented by A U B, and is consist of all the elements of A or B or Both

(Tabular method)

• A = {a, e, i, o, u} • B = {a, b, c, d, e, f} • A U B = {a, b, c, d, e, f, i, o, u}

Union operation (Selector method) A = {x:x is an even number}

B = {x:x is an odd number}

A U B = {x:x is a natural number}

P = {x:x is multiple of 5≤100}

Q = {x:x is multiple of 4≤100}

P U Q = {x:x ∈ N where x is divisible by 4 or 5}

i.e. x ∈ A U B than x ∈ A, or x ∈ B • contd........

Venn Diagram: Union operation

Properties of Union operation

A U E = E

If A ⊆ B than A U B = B

Idempotent law • A U A = A

Properties of Union operation

Commutative law • A U B = B U A

Associative law • A U (B U C) = (A U B) U C

Identity law • A U ϕ = A

• contd…….

Tabular method

A = {a,e,i,o,u}

B = {a,b,c,d,e,f}

C = {p,q,r,s}

A ∩ B = {a,e}

A ∩ C = ϕ

Intersection operation

Intersection of two sets is represented by A ∩ B and its common elements of A & B.

• i.e. any element of A ∩ B is an element of A & B both

Selector Method

A = {x:x, x is divisible by 4}

B = {x:x, x is divisible by 5}

A ∩ B = {x:x, x is divisible by 20}

Venn diagram:Intersection

Properties of Intersection

Commutative law • A ∩ B = B ∩ A

Associative law • A ∩ (B ∩ C) = (A ∩ B) ∩ C

Identity law • A ∩ E = A

Properties of Intersection

Zero prop. • A ∩ ϕ= ϕ

Idempotent law • A ∩ A = A

If A ⊆ B than A ∩ B = A

(A ∩ B) ⊆ A and (A ∩ B) ⊆ B

Common Property of Union and Intersection

Distributive law • A U (B ∩ C) = (A U B) ∩ (A U C) • A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Compliment operation

Remark :- To find compliment knowledge of universal set is compulsary • Compliment of a set is represented by A‘ or

AC • Ā or –A or ~A or U-A. • And is consist of elements which are not in A

Tabular method

A = {2,4,6,8}

E = {1,2,3…..10}

A‘ = {1,3,5,7,9,10} • (Selector method)

X ∈ A‘ => X ∉ A

Venn diagram: Complement Operation

Properties of compliment

A ∩ A’ = ϕ

A U A’ = E

E’ = ϕ and ϕ’ = E

(A’)’ = A

Properties of compliment

A ⊂ B ,=> B’ ⊂ A’

DE-MORGAN’S LAW • (A U B)’ = A’ ∩ B’ • (A ∩ B)’ = A’ U B’

Difference of sets Difference of two sets is represented by either A – B or A~ B and is consist of all the elements of A which are Not in B

(Tabular method)

A = {a,e,i,o,u}

B = {a,b,c,d,e,f}

A – B = {i,o,u}

B – A = {b,c,d,f}

Difference of sets

Selector method Venn Diagram

Properties of difference of sets

A-B ⊆ A and B-A ⊆ B

A-B, A ∩ B and B-A are mutually disjoint sets

DE-MORGAN’S LAW • A-(B U C) = (A-B) ∩ (A-C) • A-(B ∩ C) = (A-B) U (A-C)

Symmetric difference

It is Represented by A Δ B and is consist of union of A-B and B-A

i.e. A Δ B = (A-B) U (B-A)

• A = {a,e,i,o,u} • B = {a,b,c,d,e,f} • A – B = {i,o,u} • B – A = {b,c,d,f} • A Δ B = {b,c,d,f,i,o,u}

Ordered pair

A pair of two elements where first element belongs to first set and second element belongs to second set and is represented by (a, b) where a ∈ A and b ∈ B.

• Remark :- (a,b) ≠ (b,a)

Cartesian product set

If A and B are any two set than the set of all ordered pair whose first member belongs to set A and Second member belongs to set B is called the Cartesian product of A and B in that order is denoted by A X B and read as A Cross B

Cartesian product set

A = {a,b,c}

B = {p,q}

A X B = {(a,p),(a,q),(b,p),(b,q),(c,p),(c,q)}

B X A = {(p,a),(p,b),(p,c),(q,a),(q,b),(q,c)}

A X B ≠ B X A

Partition of set

Under partition of set a universal set say U is subdivided into sub sets which are disjoint but make into a union U, we can say

Number of elements in a finite set

In case of disjoint sets • n(AUB) = n(A) + n(B)

In case of joint set • n(AUB) = n(A) + n(B) – n(A ∩ B) • (AUBUC) = n(A) + n(B)+n(C)

– n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) • contd……..

MCQ’s .

MCQ.1

1. In a group of 20 children, 8 drink tea but not coffee and 13 like tea. The number of children drinking coffee but not tea is

(a) 6

(b) 7

(c) 1

(d) none of these

Answer:(B)

MCQ.2

2.If A has 32 elements, B has 42 elements and A ∪ B has 62 elements, the number of elements in A ∩ B is • (a) 12 • (b) 74 • (c) 10 • (d) none of these

Answer: A

MCQ.3

3. Given A = {2, 3}, B = {4, 5}, C = {5, 6} then A × (B ∩ C) is • (a) {(2, 5), (3, 5)} • (b) {(5, 2), (5, 3)} • (c) {(2, 3), (5, 5)} • (d) none of these

Answer:A

MCQ.4

4.In a class of 60 students, 40 students like Maths, 36 like Science, and 24 like both the subjects. Find the number of students who like • (i) Maths only. • (ii) Science only • (iii) either Maths or Science • (iv) neither Maths nor Science.

Solution

Let M = students who like Maths and S = students who like Science Then n( M) = 40, n(S) = 36 and n (M ∩ S ) = 24 • Hence, (i) n(M) – n(M ∩ S) = 40 – 24 = 16 = number of

students like Maths only. • (ii) n( S ) – n(M ∩ S) = 36 – 24 = 12 = number of students

like Science only. • (iii) n(M ∪ S) = n(M) + n(S) – n(M ∩ S) = 40 + 36 – 24 =

52 = number of students who like either Maths or Science. • ( iv) n(M ∪ S)c = 60 – n(M ∪ S ) = 60 – 52 = 8 = number

of students who like neither Maths nor Science.

MCQ.5

5. A ∩ A is equal to

(a) ϕ

(b) A

(c) E

(d) none of these

Answer:(B)

MCQ.6

A ∩ A’ is equal to • (a) ϕ • (b) A, • (c) E, • (d) none of these

Answer: A

MCQ.7

A U A’ is equal to • (a) ϕ • (b) A, • (c) E, • (d) none of these

Answer:C

MCQ.8

(A ∪ B)' is equal to • (a) (A ∩ B)' • (b) A ∪ B' • (c) A' ∩ B' • (d) none of these

Answer:C

MCQ.9

A ∩ E is equal to

(a) A

(b) E

(c) ϕ

(d) none of these

Answer:(A)

MCQ.10

If E = {1, 2, 3, 4, 5, 6, 7, 8, 9}, the subset of E satisfying 5 + x > 10 is

(a) {5, 6, 7, 8, 9}

(b) {6, 7, 8, 9}

(c) {7, 8, 9},

(d) none of these

Answer:B

MCQ.11

11. Out 2000 staff 48% preferred coffee 54% tea and 64% cocoa. Of the total 28% used coffee and tea 32% tea and cocoa and 30% coffee and cocoa. Only 6% did none of these. Find the number having all the three. • (A) 360 • (B) 280 • (C) 160 • (D) None

Answer:(A)

Thank you