Class 11 Mathematics - Sets

Exercise- 1.1:-

1)Which of the following are sets ? Justify your answer.

(i) The collection of all the months of a year beginning with the letter J.

Ans- This is a set

(ii) The collection of ten most talented writers of India.

Ans- This is not a set

(iii) A team of eleven best-cricket batsmen of the world.

Ans- This is not a set

(iv)The collection of all boys in your class.

Ans- This is a set

(v)The collection of all natural numbers less than 100.

Ans- This is a set

(vi) A collection of novels written by the writer Munshi Prem Chand.

Ans- This is a set

(vii) The collection of all even integers.

Ans- This is a set

(viii) The collection of questions in this Chapter.

Ans- This is a set

(ix) A collection of most dangerous animals of the world.

Ans- This is not a set

2)Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:

Ans:-

(i) 5 ∈ A

(ii) 8 ∉ A

(iii) 0 ∉ A

(iv) 4 ∈ A

(v) 2 ∈ A

(vi) 10 ∉ A

3)Write the following sets in roster form:

(i) A = {x : x is an integer and –3 ≤ x < 7}

Ans- A= {-3,-2,-1,0,1,2,3,4,5,6}

(ii) B = {x : x is a natural number less than 6}

Ans- B= {1,2,3,4,5}

(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}

Ans- C= {17,26,35,44,53,62,71}

(iv) D = {x : x is a prime number which is divisor of 60}

Ans- D= {2,3,5}

(v)The collection of all natural numbers less than 100.

Ans- This is a set

(vi) E = The set of all letters in the word TRIGONOMETRY

Ans- E= {T,R,I,G,O,N,M,E,Y}

(vii) F = The set of all letters in the word BETTER

Ans- F= {B,E,T,R}

4)Write the following sets in the set-builder form :

(i) {3, 6, 9, 12}

Ans- {x is a positive integer multiple of 3}

(ii) {2,4,8,16,32}

Ans- {x is 2ⁿ and 1≤n≤5 and n∈N}

(iii) {5, 25, 125, 625}

Ans- {x=5ⁿ and 1≤n≤4 and n∈N}

(iv) {2, 4, 6, . . .}

Ans- {x is an even natural number}

(v) {1,4,9, . . .,100}

Ans- {x=n²; 1≤n≤10, n∈N}

5)List all the elements of the following sets :

(i) A = {x : x is an odd natural number}

Ans- A= {1,3,5,7,9,11,13,...∞}

(ii) B = {x : x is an integer, -1/2< x < 9/2

Ans- B= {0,1,2,3,4}

(iii) C = {x : x is an integer, x²≤ 4}

Ans- C= {-2,-1,0,1,2}

(iv) D = {x : x is a letter in the word “LOYAL”}

Ans- D= {L,O,Y,A}

(v) E = {x : x is a month of a year not having 31 days}

Ans- E= {February, April, June, September, November}

(vi) F = {x : x is a consonant in the English alphabet which precedes k }

Ans- F= {b,c,d,,f,g,h,j}

6)Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

(i){1,2,3,6} --------> (c){x is an natural number divisor or 6}

(ii){2,3} --------> (a){x is a prime number & a divisor of 6}

(iii){M,A,T,H,E,I,C,S} --------> (d){x is a letter of word MATHEMATCS}

(iv){1,3,5,7,9} --------> (b){x is an odd natural number less than 10}

Exercise 1.2:-

1)Which of the following are examples of the null set

(i) Set of odd natural numbers divisible by 2

Ans- This is a null set

(ii) Set of even prime numbers

Ans- This is not a null set

(iii) { x : x is a natural numbers, x < 5 and x > 7 }

Ans- This is a null set

(iv) { y : y is a point common to any two parallel lines}

Ans- This is a null set

2)Which of the following sets are finite or infinite

(i) The set of months of a year

Ans- Finite

(ii) {1, 2, 3, . . .}

Ans- Infinite

(iii) {1, 2, 3, . . .99, 100}

Ans- Finite

(iv) The set of positive integers greater than 100

Ans- Infinite

(v) The set of prime numbers less than 99

Ans- Finite

3)State whether each of the following set is finite or infinite:

(i) The set of lines which are parallel to the x-axis

Ans- Infinite

(ii) The set of letters in the English alphabet

Ans- Finite

(iii) The set of numbers which are multiple of 5

Ans- Infinite

(iv) The set of animals living on the earth

Ans- Finite

(v) The set of circles passing through the origin (0,0)

Ans- Infinite

4)In the following, state whether A = B or not:

(i) A = { a, b, c, d }; B = { d, c, b, a }

Ans- A=B

(ii) A = { 4, 8, 12, 16 }; B = { 8, 4, 16, 18}

Ans- A≠B

(iii) A = {2, 4, 6, 8, 10}; B = { x : x is positive even integer and x ≤ 10}

Ans- A=B

(iv) A = { x : x is a multiple of 10}; B = { 10, 15, 20, 25, 30, . . . }

Ans- A≠B

5)Are the following pair of sets equal ? Give reasons.

(i) A = {2, 3}, B = {x : x is solution of x² + 5x + 6 = 0}

Ans- This sets are not equal

Reason:-

x²+5x+6=0

x²+2x+3x+6=0

x(x+2)+3(x+2)=0

x=-3, x=-2

(ii) A = { x : x is a letter in the word FOLLOW}, B = { y : y is a letter in the word WOLF}

Ans- This sets are euqal. Becuase both words only contain 'F,O,L,W'

6)From the sets given below, select equal sets :

A = { 2, 4, 8, 12}, B = { 1, 2, 3, 4}, C = { 4, 8, 12, 14}, D = { 3, 1, 4, 2}, E = {–1, 1}, F = { 0, a}, G = {1, –1}, H = { 0, 1}

Ans-

B=D

E=G

Exercise 1.3:-

1)Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :

(i) { 2, 3, 4 } ⊂ { 1, 2, 3, 4,5 }

(ii) { a, b, c } ⊄ { b, c, d }

(iii) {x : x is a student of Class XI of your school} ⊂ {x : x student of your school}

(iv) {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit}

(v) {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}

(vi) {x : x is an equilateral triangle in a plane} ⊂ {x : x is a triangle in the same plane}

(vii) {x : x is an even natural number} ⊂ {x : x is an integer}

2)Examine whether the following statements are true or false:

(i) { a, b } ⊄ { b, c, a }

Ans- False

(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}

Ans- True

(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }

Ans- False

(iv) { a } ⊂ { a, b, c }

Ans- True

(v) { a } ∈ { a, b, c }

Ans- False

(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36}

Ans- True

3)Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

(i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A

Ans- Statement (i),(v),(vii),(viii),(ix),(xi) are incorrect

4)Write down all the subsets of the following sets

(i) { a }

Ans- {a}, Φ

(ii) {a, b}

Ans- {a}, {b}, {a,b}, Φ

(iii) {1, 2, 3}

Ans- Φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

(iv) Φ

Ans- Φ

5)Write the following as intervals :

(i) {x : x ∈ R, – 4 < x ≤ 6}

Ans- (-4,6]

(ii) {x : x ∈ R, – 12 < x < –10}

Ans- (-12,-10)

(iii) {x : x ∈ R, 0 ≤ x < 7}

Ans- [0,7)

(iv) {x : x ∈ R, 3 ≤ x ≤ 4}

Ans- [3,4]

6)Write the following intervals in set-builder form :

(i) (– 3, 0)

Ans- {x∈R, -3 < x < 0}

(ii) [6 , 12]

Ans- {x∈R, 6≤x≤12}

(iii) (6, 12]

Ans- {x∈R, 6 < x ≤ 12}

(iv) [–23, 5)

Ans- {x∈R, -23≤x<5}

8)Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C

(i) {0, 1, 2, 3, 4, 5, 6}

Ans- A⊂{0,1,2,3,4,5,6}

B⊂{0,1,2,3,4,5,6}

C⊄{0,1,2,3,4,5,6}

(ii) φ

Ans- A⊄φ

B⊄φ

C⊄φ

Hence, cannot be a Universal set

(iii) {0,1,2,3,4,5,6,7,8,9,10}

Ans- A⊂{0,1,2,3,4,5,6,7,8,9,10}

B⊂{0,1,2,3,4,5,6,7,8,9,10}

C⊂{0,1,2,3,4,5,6,7,8,9,10}

Hence,it an Universal set

(iv) {1,2,3,4,5,6,7,8}

Ans- A⊂{1,2,3,4,5,6,7,8}

B⊂{1,2,3,4,5,6,7,8}

C⊄{1,2,3,4,5,6,7,8}

Hence, cannot be a universal set

Exercise 1.4:-

1)Find the union of each of the following pairs of sets :

(i) X = {1, 3, 5}; Y = {1, 2, 3}

Ans- X∪Y= {1,2,3,5}

(ii) A = [ a, e, i, o, u} B = {a, b, c}

Ans- A∪B= {a,b,c,e,i,o,u}

(iii) A = {x : x is a natural number and multiple of 3}; B = {x : x is a natural number less than 6}

Ans- A∪B= {x=1,2,4,5 or a multiple of 3}

(iv) A = {x : x is a natural number and 1 < x ≤6 }; B = {x : x is a natural number and 6 < x < 10 }

Ans- A∪B= {1 < x < 10, x∈N}

(v) A = {1, 2, 3}, B = φ

Ans- A∪B= {1,2,3}

2)Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?

Ans- Yes, A∪B= {a,b,c}

3)If A and B are two sets such that A ⊂ B, then what is A ∪ B ?

Ans- A∪B is B

4)If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

(i) A∪B

Ans- {1,2,3,4,5,6}

(ii) A∪C

Ans- {1,2,3,4,5,6,7,8}

(iii) B∪C

Ans- {3,4,5,6,7,8}

(iv) B∪D

Ans- {3,4,5,6,7,8,9,10}

(v) A∪B∪C

Ans- {1,2,3,4,5,6,7,8}

(vi) A∪B∪D

Ans- {1,2,3,4,5,6,7,8,9,10}

(vii) B∪C∪D

Ans- {3,4,5,6,7,8,9,10}

5)Find the intersection of each pair of sets of question 1 above.

(i) X∩Y= {1,3}

(ii) A∩B= {a}

(iii) {3}

(iv) Φ

(v) Φ

6)If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find

(i) A∩B

Ans- {7,9,11}

(ii) B∩C

Ans- {11,13}

(iii) A∩C∩D

Ans- Φ

(iv) A∩C

Ans- {11}

(v) B∩D

Ans- Φ

(vi) A∩(B∪C)

Ans- {7,9,11}

(vii) A∩D

Ans- Φ

(viii) A∩(B∪D)

Ans- {7,9,11}

(ix) (A∩B)∩(B∪C)

Ans- {7,9,11}

(x) (A∪D)∩(B∪C)

Ans- {7,9,11,15}

7)If A = {x : x is a natural number }, B = {x : x is an even natural number}, C = {x : x is an odd natural number}andD = {x : x is a prime number }, find

(i) A∩B

Ans- B

(ii) A∩C

Ans- C

(iii) A∩D

Ans- D

(iv) B∩C

Ans- Φ

(v) B∩D

Ans- {2}

(vi) C∩D

Ans- {x is a odd Prime number}

8)Which of the following pairs of sets are disjoint

Ans- (iii) {x : x is an even integer } and {x : x is an odd integer}

9)If A = {3, 6, 9, 12, 15, 18, 21}, B = { 4, 8, 12, 16, 20 }, C = { 2, 4, 6, 8, 10, 12, 14, 16 }, D = {5, 10, 15, 20 }; find

(i) A-B

Ans- {3,6,9,15,18,21}

(ii) A-C

Ans- {3,9,15,18,21}

(iii) A-D

Ans- {3,6,9,12,18,21}

(iv) B-A

Ans- {4,8,16,20}

(v) C-A

Ans- {2,4,8,10,14,16}

(vi) D-A

Ans- {5,10,20}

(vii) B-C

Ans- {20}

(viii) B-D

Ans- {4,8,12,16}

(ix) C-B

Ans- {2,6,10,14}

(x) D-B

Ans- {5,10,15}

(xi) C-D

Ans- {2,4,6,8,12,14,16}

(xii) D-C

Ans- {5,15,20}

10)If X= { a, b, c, d } and Y = { f, b, d, g}, find

(i) X-Y

Ans- {a,c}

(ii) Y-X

Ans- {f,g}

(iii) X∩Y

Ans- {b,d}

11)If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

Ans- Set of irrational numbers

12)State whether each of the following statement is true or false. Justify your answer.

(i) { 2, 3, 4, 5 } and { 3, 6} are disjoint sets.

Ans- False

(ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets

Ans- False

(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.

Ans- True

(iv) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.

Ans- True

Exercise 1.5:-

1)Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }.

(i) A′

Ans- {5,6,7,8,9}

(ii) B′

Ans- {1,3,5,7,9}

(iii) (A ∪ C)′

Ans- {7,8,9}

(iv) (A ∪ B)′

Ans- {5,7,9}

(v) (A′)′

Ans- {1,2,3,4}

(vi) (B – C)′

Ans- {1,3,4,5,6,7,9}

2)If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :

(i) A = {a, b, c}

Ans- {d,e,f,g,h}

(ii) B = {d, e, f, g}

Ans- {a,b,c,h}

(iii) C = {a, c, e, g}

Ans- {b,d,f,h}

(iv) D = { f, g, h, a}

Ans- {b,c,d,e}

3)If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :

(i) {x : x is an even natural number}

Ans- {x : x is an odd natural number}

(ii) { x : x is an odd natural number }

Ans- {x : x is an even natural number}

(iii) {x : x is a positive multiple of 3}

Ans- {x : x∈N and x is not a multiple of 3}

(iv) { x : x is a prime number }

Ans- { x : x is a positive composite numbers or x=1 }

(v) {x : x is a natural number divisible by 3 and 5}

Ans- {x : x is a positive integer which is not divisible by 3 and not divisible by 5}

(vi) { x : x is a perfect square }

Ans- {x : x∈N and x is not a perfect square

(vii) { x : x is a perfect cube}

Ans- {x : x∈N and x is not a perfect square}

(viii) { x : x + 5 = 8 }

Ans- {x : x∈N and x≠3}

(ix) { x : 2x + 5 = 9}

Ans- {x : x∈N and x≠2}

(x) { x : x ≥ 7 }

Ans- {x : x∈N and x < 7}

(xi) { x : x ∈ N and 2x + 1 > 10 }

Ans- {x : x∈N and x≤ 9/2}

4)If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :

(i) (A∪B)′=A′∩B′

Ans- U={1,2,3,4,5,6,7,8,9}

A={2,4,6,8}

B={2,3,5,7}

(A∪B)′={2,3,4,5,6,7,8}′={1,9}

A′∩B′={1,3,5,7,9}∩{1,4,6,8,9}={1,9}

Therefore, (A∪B)′=A′∩B′

(ii) (A∩B)′=A′∪B′

Ans- (A∩B)′={2}′={1,3,4,5,6,7,8,9}

A′∪B′={1,35,7,9}∪{1,4,6,8,9} = {1,3,4,5,6,7,8,9}

Therefore, (A∩B)′=A′∪B′

6)Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?

Ans- A′ is the set of all equilateral triangles

7)Fill in the blanks to make each of the following a true statement :

(i) A∪A′

Ans- U

(ii) Φ′∩A

Ans- A

(iii) A∩A′

Ans- Φ

(iv) U′∩A

Ans- Φ