# Question-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.

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

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

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

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

(vi) A collection of novels written by the writer munshi prem chand.

(vii) The collection of all even integers.

(viii) The Collection of questions in this chapter.

(ix)

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

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

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

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

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

(vi) A collection of novels written by the writer munshi prem chand.

(vii) The collection of all even integers.

(viii) The Collection of questions in this chapter.

(ix)

**A collection of most dangerous animals of the world**.**Solution:**

(i), (iv), (v), (vi), (vii) and (viii) are sets.

# Question-2

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

(i) 5â€¦.A (ii) 8â€¦A (iii) 0â€¦.A

(iv) 4â€¦A (v) 2â€¦A (Vi) 10â€¦.A

(i) 5â€¦.A (ii) 8â€¦A (iii) 0â€¦.A

(iv) 4â€¦A (v) 2â€¦A (Vi) 10â€¦.A

**Solution:**

(i) 5 __ âˆˆ __ A

(ii) 8 __âˆ‰ __ A

(iii) 0 __ âˆ‰__ A

(iv) 4 __âˆˆ__ A

(v) 2 __âˆˆ__ A

(vi) 10 __âˆ‰__ A

# Question-3

**Write the following sets in roster form:**

(i) A = {: is an integer and â€“3<<7}

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

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

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

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

(vi)

(i) A = {: is an integer and â€“3<<7}

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

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

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

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

(vi)

**F = The set of all letters in the word BETTER****Solution:**

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

(ii) B = {1, 2, 3, 4, 5}

(iii) C = {17, 26, 35, 44, 53, 62, 71, 80}

(iv) D = {2, 3, 5}

(v) E = {T, R, I, G, O, N, M, E, R, Y}

(vi) F = {B, E, T, R}

# Question-4

**4. Write the following sets in the set-builder form:**

(i) {3,6,9,12} (ii) {2,4,6,8,16,32} (iii) {5,25,125,625}

(iv) {2,4,6,â€¦} (v) {1,4,9,â€¦.100}

(i) {3,6,9,12} (ii) {2,4,6,8,16,32} (iii) {5,25,125,625}

(iv) {2,4,6,â€¦} (v) {1,4,9,â€¦.100}

**Solution:**

(i) A = {x : x =3n and 1 â‰¤ n â‰¤ 4}

(ii) B = {x : x = 2

^{n }and 1 â‰¤ n â‰¤ 5}

(iii) C= {x : x = 5

^{n}and 1 â‰¤ n â‰¤ 4}

(iv) D = {x : x is an even natural number}

(v) E = {x : x = n

^{2}and 1 â‰¤ n â‰¤ 10}

# Question-5

**List all the elements of the following sets:**

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

(ii) B = { X:X is an integer,- < X < }

(iii) C = { X:X is an integer, X

(iv) D = { X:X is a letter in the word " LOYAL" }

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

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

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

(ii) B = { X:X is an integer,- < X < }

(iii) C = { X:X is an integer, X

^{2}â‰¤ 4 }(iv) D = { X:X is a letter in the word " LOYAL" }

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

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

**Solution:**

(i) A = { 1, 3, 5, 7, 9, 11, 13â€¦. }

(ii) B = { 0, 1, 2, 3, 4 }

(iii) C = { -2, -1, 0, 1, 2 }

(iv) D = { L, O, Y, A }

(v) E = { February, April, June, September, November }

(vi) F = { b, c, d, f, g, h, j }

# Question-6

**Match each of the sets on the left described in the roster form with the**

same set on the right described in set builder form:

(i) {1, 2, 3, 6} (a) {x: x is a prime number and a divisor of 6}

(ii) {2, 3} (b) {x: x is an odd natural number less than 10}

(iii) {H, A, Y, R, N} (c) {x: x is a natural number and divisor of 6.}

(iv) {1, 3, 5, 7, 9} (d) {x: x is a letter of the word 'HARYANA'.}

same set on the right described in set builder form:

(i) {1, 2, 3, 6} (a) {x: x is a prime number and a divisor of 6}

(ii) {2, 3} (b) {x: x is an odd natural number less than 10}

(iii) {H, A, Y, R, N} (c) {x: x is a natural number and divisor of 6.}

(iv) {1, 3, 5, 7, 9} (d) {x: x is a letter of the word 'HARYANA'.}

**Solution:**

(i) {1, 2, 3, 6} (c) {x: x is a natural number and divisor of 6}

(ii){2, 3} (a) {x : x is a prime number and a divisor of 6}

(iii) {H, A, Y, R, N} (d) {x : x is a letter of the word 'HARYANA'.}

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

# Question-7

**Which of the following are examples of the null set?**

(i) Set of odd natural numbers divisible by 2.

(ii) Set of even prime numbers.

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

(iv) { y : y is a point common to any parallel lines }(i) Set of odd natural numbers divisible by 2.

(ii) Set of even prime numbers.

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

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

**Solution:**

(i) Null set.

(ii) It is not a null set because 2 is a even prime number.

(iii) Null set.

(iv) Null set.

# Question-8

**Which of the following sets is finite or infinite?**

(i) The set of the months of a year.

(ii) {1, 2, 3, â€¦â€¦}

(iii) {1, 2, 3, â€¦â€¦.., 99, 100}

(iv) The set of positive integers greater than 100.

(v) The set of prime numbers less that 99.

(i) The set of the months of a year.

(ii) {1, 2, 3, â€¦â€¦}

(iii) {1, 2, 3, â€¦â€¦.., 99, 100}

(iv) The set of positive integers greater than 100.

(v) The set of prime numbers less that 99.

**Solution:**

(i) Finite set

(ii) Infinite set

(iii) Finite set

(iv) Infinite set

(v) Finite set

# Question-9

**Which of the following sets is finite or infinite?**

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

(ii) The set of letters in the English alphabet.

(iii) The set of numbers which are multiples of 5.

(iv) The set of animals living on earth.

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

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

(ii) The set of letters in the English alphabet.

(iii) The set of numbers which are multiples of 5.

(iv) The set of animals living on earth.

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

**Solution:**

(i) Infinite set

(ii) Finite set

(iii) Infinite set

(iv) Finite set

(v) Infinite set

# Question-10

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

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

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

(iii) A = {2, 4, 6, 8, 10} B = {x :x is positive even integer less than10}

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

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

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

(iii) A = {2, 4, 6, 8, 10} B = {x :x is positive even integer less than10}

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

**Solution:**

(i) A = B

(ii) A â‰ B, because element 12 of set A is not present in set B and element 18 of set B is not present in set A.

(iii) A = B

(iv) A â‰ B, because set B consists of elements that are multiple of 5.

# Question-11

**Are the following pair of sets equal? Give reasons.**

(i) A = {2, 3}

B = {x: x is a solution of x

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

B = {y: y is a letter in the word WOLF}

(i) A = {2, 3}

B = {x: x is a solution of x

^{2}+ 5x + 6 = 0}(ii) A = {x: x is a letter in the word FOLLOW}

B = {y: y is a letter in the word WOLF}

**Solution:**

(i) A = {2, 3}

B = {x: x is a solution of x

^{2}+ 5x + 6 = 0} = {-2, -3}

Therefore the above pair are not equal sets.

(ii) A = {x:x is a letter in the word FOLLOW} = {F, O, L, W}

B = {y:y is a letter in the word WOLF} = {W, O, L, F}

Therefore the above pair are equal sets.

# Question-12

**From the sets given below, select equal sets and equivalent sets**

A = {0, a} B = {1, 2, 3, 4}

C = {4, 8, 12} D = {3, 1, 2, 4}

E = {1, 0} F = {8, 4, 12}

G = {1, 5, 7, 11} H = {a, b}

A = {0, a} B = {1, 2, 3, 4}

C = {4, 8, 12} D = {3, 1, 2, 4}

E = {1, 0} F = {8, 4, 12}

G = {1, 5, 7, 11} H = {a, b}

**Solution:**

Equal sets:

(i) B = D

(ii) C = F;

Equivalent sets:

(i) A, E, H;

(ii) D, G;

# Question-13

**Make correct statements by filling in the symbols âŠ‚ or âŠ„ in the blank spaces:**

(i) {2, 3, 4} â€¦â€¦. {1, 2, 3, 4, 5,}

(ii) {a, b, c}â€¦â€¦â€¦ {b, c, d}

(iii) {x : x is a student of Class 9 of your school.} â€¦â€¦..{x: x is a student of your school}

(iv) {x : x is a circle in the plane} â€¦â€¦..{x : x is a circle with radius 1}

(v) {x : x is a triangle in the plane} â€¦â€¦â€¦. { x : x is a rectangle in the plane}

(vi) {x : x is an equilateral triangle in the plane}â€¦â€¦â€¦.{ x : x is a triangle in the plane}

(vii) {x : x is an even natural number} â€¦â€¦â€¦ {x : x is an integer}

(i) {2, 3, 4} â€¦â€¦. {1, 2, 3, 4, 5,}

(ii) {a, b, c}â€¦â€¦â€¦ {b, c, d}

(iii) {x : x is a student of Class 9 of your school.} â€¦â€¦..{x: x is a student of your school}

(iv) {x : x is a circle in the plane} â€¦â€¦..{x : x is a circle with radius 1}

(v) {x : x is a triangle in the plane} â€¦â€¦â€¦. { x : x is a rectangle in the plane}

(vi) {x : x is an equilateral triangle in the plane}â€¦â€¦â€¦.{ x : x is a triangle in the plane}

(vii) {x : x is an even natural number} â€¦â€¦â€¦ {x : x is an integer}

**Solution:**

(i) {2, 3, 4} â€¦âŠ‚â€¦. {1, 2, 3, 4, 5,}

(ii) {a, b, c}â€¦âŠ„â€¦.. {b, c, d}

(iii) {x : x is a student of Class 9 of your school.} â€¦âŠ‚â€¦.{x; x is a student of your school.}

(iv) {x : x is a circle in the plane.} â€¦âŠ„â€¦ {x : x is a circle with radius 1.}

(v) {x : x is a triangle in the plane.} â€¦âŠ„â€¦ { x : x is a rectangle in the plane.}

(vi) {x : x is an equilateral triangle in the plane .}â€¦ âŠ‚â€¦â€¦ { x : x is a triangle in the plane.}

(vii) {x : x is an even natural number.} â€¦âŠ‚â€¦ {x : x is an integer.}

# Question-14

**Examine whether the following statements are true or false:**

(i) {a, b} âŠ„ {b, c, a}

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

(iii) {1, 2, 3} ÃŒ {1, 2, 3}

(iv) {a} ÃŒ {a, b, c}

(v) {a} ÃŽ {a, b, c}

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

(i) {a, b} âŠ„ {b, c, a}

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

(iii) {1, 2, 3} ÃŒ {1, 2, 3}

(iv) {a} ÃŒ {a, b, c}

(v) {a} ÃŽ {a, b, c}

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

**Solution:**

(i) False, because elements a and b are present in that set.

(ii) True

(iii) True

(iv) True

(v) False

(vi) True

# Question-15

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

(i) {3, 4} âŠ‚ A

(ii) {3, 4} âˆˆ A

(iii) {{3, 4}} âŠ‚ A

(iv) 1 âˆˆ A

(v) 1 âŠ‚ A

(vi) {1, 2, 5} âŠ‚A

(vii) {1, 2, 5} âˆˆ A

(viii) {1, 2, 3}âŠ‚ A

(ix) Ï† âˆˆ A

(x) Ï† âŠ‚ A

(xi) {Ï† A(i) {3, 4} âŠ‚ A

(ii) {3, 4} âˆˆ A

(iii) {{3, 4}} âŠ‚ A

(iv) 1 âˆˆ A

(v) 1 âŠ‚ A

(vi) {1, 2, 5} âŠ‚A

(vii) {1, 2, 5} âˆˆ A

(viii) {1, 2, 3}âŠ‚ A

(ix) Ï† âˆˆ A

(x) Ï† âŠ‚ A

(xi) {Ï† A

**Solution:**

(i) False, {3,4} is an element not a set.

(ii) True

(iii) True

(iv) True

(v) False, 1 is an element not a set.

(vi) True

(vii) False, {1, 2, 5} is a set not an element.

(viii) False, 3 is an element of set contained in A.

(ix) False, Ï†is not an element of A.

(x) False, Ï† is not an element of A.

(xi) True.

# Question-16

**Write down all the subsets of the following sets**

**(i) {a} (ii) { a,b} (iii) {1,2,3} (iv)**

**Solution:**

(i) {a} = {a} , { }

(ii) { a, b} = {a} , { b} , {a,b} , { }

(iii) {1,2,3} = {1}, {2}, {3} , {1,2} , {2,3} , {3,1},{1,2,3} , { }

(iv)

**=**{ }

# Question-17

**write the following intervals in set-builder form:**

**(i) {: R, -4 < 6}**

(ii) {: R, -12 < -10}

(iii) {: R, 0 <7}

(iv) {: R, 3 4}

(ii) {: R, -12 < -10}

(iii) {: R, 0 <7}

(iv) {: R, 3 4}

**Solution:**

(i) (-4,6)

(ii) (-12,-10)

(iii) (0,6)

(iv) (3,4)

# Question-18

**Write the following intervals in set-builder form:**

(i) (-3,0) (ii) [6,12] (iii) (6,12] (iv) [-23,5)(i) (-3,0) (ii) [6,12] (iii) (6,12] (iv) [-23,5)

**Solution:**

(i) (-3,0) = {x : x R â€“3 < x < 0}

(ii) [6,12] = {x : x R 6 < x 12}

(iii) (6,12] = {x : x R 6 < x 12}

(iv) [-23,5) = {x : x R -23 x < 5}

# Question-19

**What universal set(s) would you propose for each of the following:**

(i) The set of right triangles. (ii) The set of isosceles triangles.

(i) The set of right triangles. (ii) The set of isosceles triangles.

**Solution:**

(i) The set of triangle in a plane.

(ii)

**The set of triangle in a plane.**

# Question-20

**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

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}**

(ii)

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

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

(ii)

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

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

**Solution:**

c. {0,1,2,3,4,5,6,7,8,9,10}

# Question-21

**Find the following sets find their union:**

(i) A = {a,e,i,o,u}; B = {a,b,c}.

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

(iii) A = {x: x is a natural number and multiple of 3.};

B = {x : x is a natural number less than 6.}.

(iv) A = {x: x is a natural number and 1 < x

B = {x : x is a natural number and 6 < x < 10.}.

(v) A = {1,2,3}; B = Ï†.

(i) A = {a,e,i,o,u}; B = {a,b,c}.

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

(iii) A = {x: x is a natural number and multiple of 3.};

B = {x : x is a natural number less than 6.}.

(iv) A = {x: x is a natural number and 1 < x

__<__6.};B = {x : x is a natural number and 6 < x < 10.}.

(v) A = {1,2,3}; B = Ï†.

**Solution:**

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

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

(iii) A = { 3, 6, 9, 12,â€¦â€¦â€¦ }

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

A U B = { 1, 2, 3, 4, 5, 6, 9, 12................. }

(iv) A = { 2, 3, 4, 5, 6 }

B = { 7, 8, 9 }

A U B = { 2, 3, 4, 5, 6, 7, 8, 9 }

A U B = { x : 1 < x < 10, x âˆˆN }

(v) A U B = { 1, 2, 3 }

# Question-22

**Let A = {a, b}, B = {a,b, c}. Is A âŠ‚ What is A U B?**

**Solution:**

Yes, A U B = {a, b, c}

# Question-23

**If A and B are two sets such that A âŠ‚ then what is A U B?**

**Solution:**

A U B = B.

# Question-24

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

(i) AB (ii) AC (iii) BC (iv) B D

(v) ABC (vi) ABD (vii) BC D

(i) AB (ii) AC (iii) BC (iv) B D

(v) ABC (vi) ABD (vii) BC D

**Solution:**

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

(ii) AC = {1,2,3,4,5,6,7,8}

(iii) BC = {3,4,5,6,7,8}

(iv) B D = {3,4,5,6,7,8,9,10}

(v) ABC

AB = {1,2,3,4,5,6}

C = {5,6,7,8}

ABC = {1,2,3,4,5,6,7,8}

(vi) ABD

AB = {1,2,3,4,5,6}

D = {7,8,9,10}

ABD = {1,2,3,4,5,6,7,8,9,10}

(vii) BC D

BC = {3,4,5,6,7,8}

D = {7,8,9,10}

BC D = {3,4,5,6,7,8,9,10}

# Question-25

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

(i) A âˆ© B

(ii) B âˆ© C

(iii) A âˆ© C âˆ© B

(iv) A âˆ© C

(v) B âˆ© D

(vi) A âˆ© (B âˆ© C)

(vii) A âˆ© D

(viii) A âˆ© (B âˆ© D)

(ix) (A âˆ© B) âˆ© (B âˆª C)

(x) (A âˆª B) âˆ© (B âˆª C)

(i) A âˆ© B

(ii) B âˆ© C

(iii) A âˆ© C âˆ© B

(iv) A âˆ© C

(v) B âˆ© D

(vi) A âˆ© (B âˆ© C)

(vii) A âˆ© D

(viii) A âˆ© (B âˆ© D)

(ix) (A âˆ© B) âˆ© (B âˆª C)

(x) (A âˆª B) âˆ© (B âˆª C)

**Solution:**

(i) A âˆ© B = {7, 9, 11}

(ii) B âˆ© C = {11, 13}

(iii) A âˆ© C âˆ© B = {11}

(iv) A âˆ© C = {11}

(v) B âˆ© D = {}

(vi) A âˆ© (B âˆ© C) = {11}

(vii) A âˆ© D = {}

(viii) A âˆ© (B âˆ© D) = {3, 5, 7, 9, 11} âˆ© {} = {}

(ix) (A âˆ© B) âˆ© (B âˆª C) = {7, 9, 11} âˆ© {7, 9, 11, 13, 15} = {7, 9, 11}

(x) (A âˆª B) âˆ© (B âˆª C) = {3, 5, 7, 9, 11, 13} âˆ© {7, 9, 11, 13, 15}

= {7, 9, 11, 13}

# Question-26

**Let A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number }, D = {x : x is a prime number}. Find**

(i) A âˆ© B

(ii) A âˆ© C

(iii) A âˆ© D

(iv) B âˆ© C

(v) B âˆ© D

(vi) C âˆ©D

**Solution:**

A = {x : x is a natural number} = {1, 2, 3, 4, â€¦..}

B = {x : x is an even natural number} = {2, 4, 6, 8, â€¦â€¦.}

C = {x : x is an odd natural number} = {1, 3, 5, 7, 9, â€¦â€¦.}

D = {x : x is a prime number} = {2, 3, 5, 7, â€¦â€¦â€¦}

(i) A âˆ© B = {2, 4, 6, 8, â€¦â€¦.} = B

(ii) A âˆ© C = {1, 3, 5, 7, 9, â€¦â€¦.} = C

(iii) A âˆ© D = {2, 3, 5, 7, â€¦â€¦â€¦} = D

(iv) B âˆ© C = {}

(v) B âˆ©D = {2}

(vi) C âˆ© D = {3, 5, 7, â€¦â€¦â€¦} = {x : x is an odd prime number}

# Question-27

**Which of the following pairs of sets and disjoint**

(i) {1, 2, 3, 4} and {X : X is a natural number and 4 â‰¤ X â‰¤ 6}

(ii) {a, e, i, o, u} and {c, d, e, f}

(iii) {X : X is an even integer} and {X : X is an Odd integer}(i) {1, 2, 3, 4} and {X : X is a natural number and 4 â‰¤ X â‰¤ 6}

(ii) {a, e, i, o, u} and {c, d, e, f}

(iii) {X : X is an even integer} and {X : X is an Odd integer}

**Solution:**

(iv) {: is an even integer} and {: is an Odd integer}

# Question-28

**Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find**

(i) A â€“ B

(ii) A â€“ C

(iii) A â€“ D

(iv) B â€“ A

(v) C â€“ A

(vi) D â€“ A

(vii) B â€“ C

(viii) B â€“ D

(ix) C â€“ B

(x) D â€“ B

(xi) C â€“ D

(xii) D â€“ C

(i) A â€“ B

(ii) A â€“ C

(iii) A â€“ D

(iv) B â€“ A

(v) C â€“ A

(vi) D â€“ A

(vii) B â€“ C

(viii) B â€“ D

(ix) C â€“ B

(x) D â€“ B

(xi) C â€“ D

(xii) D â€“ C

**Solution:**

A = {3, 6, 12, 15, 18, 21},

B = {4, 8, 12, 16, 20},

C = {2, 4, 6, 8, 10, 12, 14, 16}

and D = {5, 10, 15, 20}.

(i) A â€“ B = {3, 6, 15, 18, 21}

(ii) A â€“ C = {3, 15, 18, 21}

(iii) A â€“ D = {3, 6, 12, 18, 21}

(iv) B â€“ A = {4, 8, 16, 20}

(v) C â€“ A = {2, 4, 8, 10, 14, 16}

(vi) D â€“ A = {5, 10, 20}

(vii) B â€“ C = {20}

(viii) B â€“ D = {4, 8, 12, 16}

(ix) C â€“ B = {2, 6, 10, 14}

(x) D â€“ B = {5, 10, 15}

(xi) C â€“ D = {2, 4, 6, 8, 12, 14, 16}

(xii) D â€“ C = {5, 15, 20}

# Question-29

**If X = {a, b, c, d} and Y = {f, b, d, g}, find (i) X â€“ Y (ii) Y â€“ X (iii) X**

**âˆ© Y**

**Solution:**

(i) X â€“ Y = {a, c}

(ii) Y â€“ X = {f, g}

(iii) X âˆ© Y = {b, d}

# Question-30

**If R is the set of real numbers and Q is the set of rational numbers, then what is R â€“ Q?**

**Solution:**

R â€“ Q is the set of all irrational numbers.

# Question-31

**State whether each of the following statements is true or false. Justify your answer.**

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

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

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

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

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

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

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

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

**Solution:**

(i) False, 3 is an element of both the sets.

(ii) False, a is an element of both the sets.

(iii) True

(iv) True

# Question-32

**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}.**

Find

(i) Aâ€™

(ii) Bâ€™

(iii) (A âˆ© C)â€™

(iv) (A âˆª B)â€™

(v) (Aâ€˜)â€™

(vi) (B - C)â€™

Find

(i) Aâ€™

(ii) Bâ€™

(iii) (A âˆ© C)â€™

(iv) (A âˆª B)â€™

(v) (Aâ€˜)â€™

(vi) (B - C)â€™

**Solution:**

(i) Aâ€™ = {5, 6, 7, 8, 9}

(ii) Bâ€™ = {1, 3, 5, 7, 9}

(iii) (A âˆ© C) = {3, 4}

(A âˆ© C)â€™ = {1, 2, 5, 6, 7, 8, 9}

(iv) (A âˆª B) = {1, 2, 3, 4, 6, 8}

(A âˆª B)â€™ = {5, 7, 9}

(v) (Aâ€˜) = {5, 6, 7, 8, 9}

(Aâ€˜)' = {1, 2, 3, 4}

(vi) (B - C) = {2, 8}

(B - C)â€™ = {1, 3, 4, 5, 6, 7, 9}

# Question-33

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

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

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

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

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

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

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

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

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

**Solution:**

Aâ€™ = {d, e, f, g, h}

Bâ€™ = {a, b, c, h}

Câ€™ = {b, d, f, h}

Dâ€™ = {b, c, d, e}

# Question-34

**Taking the set of natural numbers as the universal set, write down the complements of the following sets**

(i) {x : x âˆˆ N and x is even}

(ii) {x : x âˆˆ N and x is odd}

(iii) {x : x âˆˆ N and x = 3n for some n âˆˆ N}

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

(v) {x : x âˆˆ N and x is a perfect square}

(vi) {x : x âˆˆ N and x is a perfect cube}

(vii) {x : x âˆˆ N and x + 5 = 8}

(viii) {x : x âˆˆ N and 2x + 5 = 9}

(ix) {x : x âˆˆN and x â‰¥ 7}

(x) {x : x âˆˆ N and x is a divisible by 3 and 5}(i) {x : x âˆˆ N and x is even}

(ii) {x : x âˆˆ N and x is odd}

(iii) {x : x âˆˆ N and x = 3n for some n âˆˆ N}

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

(v) {x : x âˆˆ N and x is a perfect square}

(vi) {x : x âˆˆ N and x is a perfect cube}

(vii) {x : x âˆˆ N and x + 5 = 8}

(viii) {x : x âˆˆ N and 2x + 5 = 9}

(ix) {x : x âˆˆN and x â‰¥ 7}

(x) {x : x âˆˆ N and x is a divisible by 3 and 5}

**Solution:**

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

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

(iii) {x: x âˆˆ N and x is not a multiple of 3}

(iv) {x: x is a positive composite number and x = 1}

(v) {x: x âˆˆ N and x is not a perfect square}

(vi) {x: x âˆˆ N and x is not a perfect cube}

(vii) {x: x âˆˆ N and x â‰ 3}

(viii) {x: x âˆˆ N and x â‰ 2}

(ix) {1, 2, 3, 4, 5, 6}

(x) {x: x âˆˆ N and x is not divisible by 3 and 5}

# Question-35

**Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that**

(i) (A âˆª B)â€™ = Aâ€™ âˆ© Bâ€™

(ii) (A âˆ© B)â€™ = Aâ€™ âˆª Bâ€™

(i) (A âˆª B)â€™ = Aâ€™ âˆ© Bâ€™

(ii) (A âˆ© B)â€™ = Aâ€™ âˆª Bâ€™

**Solution:**

(i) A âˆª B = {2, 3, 4, 5, 6, 7, 8}

(A âˆª B)â€™

**= {1, 9}**

Aâ€™ âˆ© Bâ€™ = {1, 3, 5, 7, 9} âˆ© {1, 4, 6, 8, 9} = {1, 9}

Therefore (A âˆª B)â€™ = Aâ€™ âˆ© Bâ€™

(ii) A âˆ© B = {2}

(A âˆ© B)â€™ = {1, 3, 4, 5, 6, 7, 8, 9}

Aâ€™ âˆª Bâ€™ = {1, 3, 5, 7, 9} âˆª {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}

Therefore (A âˆ© B)â€™ = Aâ€™ âˆª Bâ€™

# Question-36

**Draw appropriate Venn diagram for each of the following:**

(i) (AB)

(i) (AB)

^{â€™}, (ii) A^{â€™}B^{â€™}, (iii) (AB)^{â€™}, (iv) A^{â€™}B^{â€™}**Solution:**

(i) A âˆªB ( A âˆªB)^{â€™}

** (ii) A ^{â€™} B^{â€™}**

**A ^{â€™} **

**âˆ© B**

^{â€™}

** (iii) (A****âˆ© B)'
(iv) A' âˆª B'**

# Question-37

**If X and Y are**

**two sets such that n(X) = 17, n(Y) = 23, n(X****âˆª Y) = 38, find n(X âˆ© Y).****Solution:**

We have

n(X) = 17

n(Y) = 23

n(X âˆª Y) = 38

By using the formula,

n(X âˆª Y) = n(X) + n(Y) - n(X âˆ©Y)

We have 38 = 17 + 23 - n(X âˆ© Y)

.

^{.}. n(X âˆ© Y) = 17 + 23 - 38

.

^{.}. n(X âˆ© Y) = 40 - 38

.

^{.}. n(X âˆ© Y) = 2

# Question-38

**If X and Y are two sets such that X has 8 elements, Y has 15 elements, and XâˆªY has 18 elements, how many elements does X âˆ© Y have?****Solution:**

We have n(X) = 8 n(Y) = 15

n(X âˆª Y) = 18

By using the formula,

n(X âˆª Y) = n(X) + n(Y) - n(X âˆ© Y)

We have 18 = 8 + 15 - n(X âˆ© Y)

.

^{.}. n(X âˆ© Y) = 23 -18

.

^{.}. n(X âˆ© Y) = 5.

# Question-39

**In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many can speak both Hindi and English?**

**Solution:**

Let H denote the set of people who speak Hindi and E denote the set of people who speak English. Then H âˆª E is the set of people who speak Hindi or English and H âˆ© E is the set of people who speak both.

Then n(H) = 250, n(E) = 200 and n(H âˆª E) = 400

We know that

n(H âˆª E) = n( H ) + n(E) - n(H âˆ© E)

.

^{.}. 400 = 250 + 200 - n(H âˆ© E)

n(H âˆ© E) = 450 - 400

n(H âˆ© E) = 50

.

^{.}. The number of people who speak both Hindi and English are 50.

# Question-40

**If S and T are two sets such that S has 21 elements, T has 32 elements, and S âˆ©T has 11 elements, how many elements does S âˆª T have?****Solution:**

We have n(S) = 21

n(T) = 32

n(S âˆ© T) = 11

By using the formula,

n(S âˆª T) = n(S) + n(T) - n(S âˆ© T)

We have n(S âˆª T) = 21 + 32 - 11

.

^{.}. n(S âˆª T) = 53 - 11

.

^{.}. n(S âˆª T) = 42.

# Question-41

**If X and Y are two sets such that X has 40 elements, X âˆª Y has 60 elements and X âˆ© Y has 10 elements, how many elements does Y have?**

**Solution:**

We have n(X) = 40

n(X âˆª Y) = 60

n(X âˆ© Y) = 10

By using the formula,

n(X âˆª Y) = n(X) + n(Y) - n(X âˆ© Y)

We have 60 = 40 + n(Y) - 10

n(Y) = 60 - 40 + 10

n(Y) = 60 - 30

.

^{.}. n(Y) = 30.

# Question-42

**In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?**

**Solution:**

Let C denote the set of people who like coffee and T denote the set of people who like tea. Then C âˆª T is the set of people who like at least one of the two drinks and C âˆ© T is the set of people who like both of the two drinks.

Then n(C) = 37, n(T) = 52 and n(C âˆª T) = 70

**Method 1:**

We know that

n(C âˆª T) = n(C) + n(T) - n(C âˆ© T)

.

^{.}. 70 = 37 + 52 - n(C âˆ© T)

n(C âˆ© T) = 37 + 52 - 70

n(C âˆ© T) = 89 - 70

n(C âˆ© T) = 19

.

^{.}. The number of people who like both coffee and tea are 19.

**Method 2:**

Let n(C âˆ© T) = x

From the diagram we get

n(C âˆª T) = 37 - x + x + 52 â€“ x

n(C âˆª T) = 89 - x

70 = 89 - x

.^{.}. x = 19.

.^{.}. The number of people who like both coffee and tea are 19.

# Question-43

**In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?**

**Solution:**

Let C denote the set of people who like cricket and T denote the set of people who like tennis. Then C âˆª T is the set of people who like at least one of the two games and C âˆ© T is the set of people who like both cricket and tennis.

Then n(C) = 40, n(C âˆª T) = 65 and n(C âˆ© T) = 10

**Method 1:**

We know,

n(C âˆª T) = n(C) + n(T) - n(C âˆ© T)

65 = 40 + n(T) - 10

n(T) = 65 - 40 + 10

.

^{.}. n(T) = 35

Therefore, number of people who like tennis = n(T) = 35

and the number people who like only tennis and not cricket = n(T) - n(C âˆ© T) = 35 - 10 = 25

# Question-44

**In a committee 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?**

**Solution:**

Let F denote the set of people who speak French and S denote the set of people who speak Spanish. Then F âˆª S denote the people who like to speak at least one of these two languages and F âˆ© S denote the people who like to speak both Spanish and French.

Then n(F) = 50, n(S) = 20 and n(F âˆ© S) = 10

**Method 1:**

We know,

n(F âˆª S) = n( F ) + n(S) - n(F âˆ© S)

n(F âˆª S) = 50 + 20 - 10

.

^{.}. n(F âˆª S) = 60

Therefore, 60 people like to speak at least one of these two languages.

**Method 2:**

Let n(F âˆª S) = x

From the diagram we get

n(F âˆª S) = 40 + 10 + 10

.^{.}. n(F âˆª S) = 60

Therefore, 60 people like to speak at least one of these two languages