# Question-1

**Which of the following are sets?**

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

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

** (ii) The collection of 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) The collection of novels written by the river Prem Chand. **

** (vii) The collection of all even integers. **

** (viii) The collection of different problems 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

**A**

**âˆª**

**B = {1,2,3,4,5,6,7,8,9}**

A âˆ© B {1,2} and A = {1, 2, 3, 4, 5} find the set B.

A âˆ© B {1,2} and A = {1, 2, 3, 4, 5} find the set B.

**Solution:**

A âˆª B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A âˆ© B = {1, 2} and

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

âˆ´ B = {1, 2, 6, 7, 8, 9}

# Question-3

**If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, find the complements of the following sets:**

(i) A = {2, 4, 6, 8}

(i) A = {2, 4, 6, 8}

** (ii) B = {1, 3, 5, 7, 9}
(iii) C = { 2, 3, 5, 7}
(iv) Ï† **

** (v) U **

**Solution:**

(i) A' = {1, 3, 5, 7, 9}

(ii) B' = {2, 4, 6, 8}

(iii) C' = {1, 4, 6, 8, 9}

(iv) Ï†' = U

(v) U' = Ï†

# Question-4

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

(i) 5 ______ A

(i) 5 ______ A

** (ii) 8 ______ A **

** (iii) 0 ______ A **

** (iv) 4 ______A **

** (v) 2 ______ A **

** (vi) 10 ______ 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-5

**If A âˆª B ={2, 3, 4, 5, 6, 8, 9, 11}, A âˆ© B {5,8} and B = {2, 5, 8, 9} find the set A â€“ B**

**Solution:**

A âˆª B ={2, 3, 4, 5, 6, 8, 9, 11},

A âˆ© B = {5,8}

B = {2, 5, 8, 9}

âˆ´ A = {3, 4, 5, 6, 8, 11}

**âˆ´ **A â€“ B = {3, 4, 6, 11}

# Question-6

**If U is the set of all natural numbers and A' is the set of all composite numbers, what is A?**

**Solution:**

U = {1, 2, 3, 4, 5, 6, â€¦â€¦â€¦â€¦â€¦â€¦.}

A' = {4, 6,â€¦â€¦â€¦â€¦..}

Then A = {1, 2, 3, 5, â€¦â€¦â€¦â€¦..}

# Question-7

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

(i) A = {x : x is an integer and â€“3 â‰¤ x < 7}

(i) A = {x : x is an integer and â€“3 â‰¤ x < 7}

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

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

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

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

** (vi) F = the set of all letters in the word SETS. **

**Solution:**

(i) A = {-3, -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}

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

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

(vi) F = {S, E, T}

# Question-8

**If A = {p, q, r, s} find A âˆ© A and A âˆª A.**

**Solution:**

A = {p, q, r, s}

âˆ´ A âˆ© A = {p, q, r, s}

âˆ´ A âˆª A = {p, q, r, s}

# Question-9

**Which of the following statements are true and which are false?**

** (i) U' = Ï† **

** (ii) Ï† ' = U **

** (iii) For any two subsets, X and Y of U,
(X âˆª Y)' = X' âˆª Y'**

** (iv) For any two subsets, X and Y of U,
(X âˆ© Y)' = X' âˆ© Y'**

** (v) For any two subsets, S and T of U,
(S âˆª T)' = S' âˆ© T'**

** (vi) For any two subsets S and T of U,
(S âˆ© T)' = S' âˆª T'**

**Solution:**

(i) True

(ii) True

(iii) False

(iv) False

(v) True

(vi) True

# Question-10

**If A = {x : x is a letter in the word, â€˜followâ€™} and**

B = {x : x is a letter in the word, â€˜wolfâ€™}, show that A = B.

B = {x : x is a letter in the word, â€˜wolfâ€™}, show that A = B.

**Solution:**

Clearly, A = {f, o , l, w} and B = {w, o, l, f}.

Since every element of A is in B and every element of B is in A, so A = B.

# Question-11

**Express the following sets by using the set builder method:**

** (i) A = { 1, 3, 5, 7, 9}
(ii) B = {2, 4, 6, 8}
(iii) C = {-1, 1}
(iv) D = {1, 5, 10, 15, â€¦.}
(v) E = {14, 21, 28, 35, 42, â€¦..,98}**

**Solution:**

(i) A = { x : x is an odd natural number, x Â£ 9}

(ii) B = { x : x is an even natural number, x â‰¤ 8}

(iii) C= {x : x is an odd natural number and |x| < 2}

(iv) D = {x : x is a natural number multiple of 5 and x = 1}

(v) E = {x : x is a multiple of 7 and 7<x<100}

# Question-12

**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**

^{0}, what is A?**Solution:**

U = The set of all triangles in a plane.

A = The set of all triangles with at least one angle different from 60^{0}.

A' = The set of all equilateral triangles.

# Question-13

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

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

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

(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, -1/2<x<9/2}

(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, 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-14

**Represent the following sets in a Venn diagram: U = {2, 3, 5, 7, 11}, A = {2, 3}**

**Solution:**

# Question-15

**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'.}

(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-16

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

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

(ii) {1, 2, 3, â€¦â€¦}

(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-17

**Represent the following sets in a Venn diagram:**

** U = { x : x is a natural number and 2 â‰¤ x â‰¤ 8.} **

** A = {x : x âˆˆ U and x divides 18} **

** B = {x: x âˆˆ U and x is a prime divisor of 18} **

**Solution:**

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

A = {2, 3, 6}

and B = {2, 3}

# Question-18

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

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

(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. **

**Solution:**

(i) Infinite set

(ii) Finite set

(iii) Infinite set

(iv) Finite set

(v) Infinite set

# Question-19

**If A, B and C are three subsets of the universal set U, draw a venn diagram showing A âˆª (B âˆª C)**

**Solution:**

# Question-20

**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} **

**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-21

**If A, B and C are three subsets of the universal set U, draw a venn diagram showing (A âˆ© B) âˆ© C.**

**Solution:**

# Question-22

**If A, B and C are three subsets of the universal set U, draw a venn diagram showing [(A âˆª B) âˆª C]'**

**Solution:**

# Question-23

**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 than 10}

(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 than 10}

(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-24

**If A, B and C are three subsets of the universal set U, draw a venn diagram showing (A' âˆ© B' ) âˆ© C'**

**Solution:**

# Question-25

**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 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-26

**If A, B, and C are three subsets of the universal set U, draw Venn diagrams for the following:B âˆ© C, when B âŠ‚ C.**

**Solution:**

# Question-27

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

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

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-28

**Given that A = {6, 7, 8, 9, 10} and B = {2, 3, 4, 5}. Write down all ordered pairs (a, b) such that a is divisible by b and hence write down the set ordered pairs given the relation â€˜is a multiple ofâ€™ from A and B.**

**Solution:**

A = {6, 7, 8, 9, 10} and B = {2, 3, 4, 5}

List of all ordered pairs : (6, 2), (6, 3), (6, 4), (6, 5), (7, 2), (7, 3), (7, 4), (7, 5), (8, 2), (8, 3), (8, 4), (8, 5), (9, 2), (9, 3), (9, 4), (9, 5), (10, 2), (10, 3), (10, 4), (10, 5)

The ordered pairs (a, b) such that a is divisible by b: (6, 2), (6, 3), (8, 2), (8, 4), (9, 3), (10, 2), (10,5)

# Question-29

**Which of the following statements are true?**

(i) The set of all cats is contained in the set of all animals.

(i) The set of all cats is contained in the set of all animals.

** (ii) The set of all isosceles triangles is contained in the set of all equilateral triangles. **

** (iii) The set of all rectangles is contained in the set of all squares. **

** (iv) The sets A = {1} and B = {{1}} are equal. **

** (v) The sets A = {x : x is a letter in the word "TITLE"} and B = {x : x is a letter in the word "LITTLE"} are equal.**

**Solution:**

(i) True

(ii) False

(iii) False

(iv) False

(v) True

# Question-30

**Let R be the relation defined by â€˜divides" from A = {2, 3, 5} to B = {6, 10, 12, 15}. Represent R (i) as a set of ordered pairs (ii) as a graph and (iii) by an arrow diagram.**

**Solution:**

A = {2, 3, 5} to B = {6, 10, 12, 15}

(i) R = {(2, 6), (2, 10), (2, 12), (2, 15), (3, 6), (3, 10), (3, 12), (3, 15), (5, 6), (5, 10), (5, 12), (5, 15)}

(ii)

(iii)

# Question-31

**A and C are disjoint sets and both A and C are subsets of B, draw Venn diagrams for the following.**

**Solution:**

# Question-32

**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-33

**If A and B are two sets such that A has 21 elements, B has 17 elements, and A âˆª B has 21 elements, how many elements does A âˆ© B have?**

**Solution:**

We have n(A) = 12

n(B) = 17

n(A âˆª B) = 21

By using the formula,

n(A âˆª B) = n(A) + n(B) - n(A âˆ© B)

We have 21 = 12 + 17 - n(A âˆ© B)

âˆ´ n(A âˆ© B) = 29 -21

âˆ´ n(A âˆ© B) = 8

# Question-34

**If P = {2, 3, 4, 8, 9} write down the sets of ordered pairs representing the relations (i) is a factor of (ii) is divisible by (iii) is a multiple of 2 on P.**

**Solution:**

P = {2, 3, 4, 8, 9}

(i) The sets of ordered pairs representing "is a factor of" : {(2, 2), (2, 4), (2, 8), (3, 3), (3, 9), (4, 4), (4, 8), (8, 8), (9, 9)}

(ii) The sets of ordered pairs representing "is divisible by" : {(2, 2), (4, 2), (8, 2), (4, 4), (8, 4), (8, 8), (9, 3), (9, 9), (3, 3)}

(iii) The sets of ordered pairs representing "is a multiple of 2 on P" : {((2, 2), (4, 2)), (8, 2)}

# Question-35

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

(i) {a, b} âŠ„ {b, c, a}

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

(iii) {1, 2, 3} âŠ‚ {1, 2, 3}

(iv) {a} âŠ‚ {a, b, c}

(v) {a} âˆˆ {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} âŠ‚ { x : x is a vowel in the English alphabet.}

(iii) {1, 2, 3} âŠ‚ {1, 2, 3}

(iv) {a} âŠ‚ {a, b, c}

(v) {a} âˆˆ {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-36

**Describe the relation R defined from A to B where A = {- 1, 2, 3, 4} to B = {-2, 4, 6} by the set R = {(- 1, - 2), (2, 4), (3, 6)}.**

**Solution:**

A = {- 1, 2, 3, 4} to B = {-2, 4, 6}

set R = {(- 1, - 2), (2, 4), (3, 6)} is the relation "is half of"

# Question-37

**If A and B are disjoint sets, show that n(A âˆª B) = n(A) + n(B)**

**Solution:**

We know that, if A and B are disjoint sets then n(A âˆ© B) = Ï† .

Hence, by using the formula n(A âˆª B) = n(A) + n(B) - n(A âˆ© B)

We have n(A âˆª B) = n(A) + n(B) - Ï†

âˆ´ n(A âˆª B) = n(A) + n(B)

Example: Let A = {1, 2} and B = {3, 4},

then A âˆª B = {1, 2, 3, 4} and A âˆ© B = Ï†

Now n(A) = 2, n(B) = 2, n(A âˆª B) = 4 and n(A âˆ© B) = Ï†

Hence, n(A âˆª B) = n(A) + n(B)

# Question-38

**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

(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

**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.

# Question-39

**Write the power set of A = {3,6,9}.**

**Solution:**

P(A)= {Ï† , {3}, {9},{6},{3,6},{3,9},{6,9},{3,6,9}}.

# Question-40

**Describe the relation, domain and range if (i) R = {(1, 1), (8, 2), (27, 3), (64, 4)} (ii) R = {(Delhi, India), (Paris, France), (Karachi, Pakistan)} (iii) R = {(4, - 2), (9, - 3), (1, 1), (4, 2), (1, - 1), (9, 3)}**

**Solution:**

(i) R = {(1, 1), (8, 2), (27, 3), (64, 4)}

R is the relation "is the cube of "

Domain = {1, 8, 27, 64}

Range = {1, 2, 3, 4}

(ii) R = {(Delhi, India), (Paris, France), (Karachi, Pakistan)}

R is the relation "is the capital of"

Domain = {Delhi, Paris, Karachi}

Range = {India , France, Pakistan}

(iii) R = {(4, - 2), (9, - 3), (1, 1), (4, 2), (1, - 1), (9, 3)}

R is the relation "is the square of"

Domain = {1, 4, 9}

Range = (- 3, - 2, - 1, 2 , 3}

# Question-41

**If x âˆˆ {1, 3, 7}, y âˆˆ {0, 2, 8} and R is the relation such that x + y < 8, represent R (i) as a set of ordered pairs and (ii) by an arrow diagram.**

**Solution:**

(i) x âˆˆ {1, 3, 7}, y âˆˆ {0, 2, 8} , R is such that x + y < 8.

The set of ordered pairs = {(1, 0), (1, 2), (3, 0), (3, 2), (7, 0)}

(ii)

# Question-42

**Which of the following sets are equal ?**

A = {x : x âˆˆ N, x < 3}, B = {1, 2}, C = {3, 1}

D = {x : x âˆˆ N, x is odd, x < 5}, E = {1, 2, 1}, F = {1, 1, 3}

A = {x : x âˆˆ N, x < 3}, B = {1, 2}, C = {3, 1}

D = {x : x âˆˆ N, x is odd, x < 5}, E = {1, 2, 1}, F = {1, 1, 3}

**Solution:**

A = {1, 2}, B = {1, 2}, C = {3, 1},

D = {1, 3}, E = {1, 2, 1}, F = {1, 1, 3}

A, B, E and C, D, F are equal sets.

# Question-43

**If A and B are two sets such that Aâˆ¨ B has 25 elements, A has 10 elements, and B has 37 elements, how many elements does A âˆ§ B have?**

**Solution:**

n(A âˆª B) = 25; n(A) = 10 ; n(B) = 37

n(A âˆª B) = n(A) + n(B) â€“ n(Aâˆ© B)

âˆ´ 25 = 10 + 37 - n(A âˆ© B)

â‡’ n(A âˆ© B) = 12

**âˆ´ **Aâˆ§ B has 12 elements.

# Question-44

**In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea. Find :**

(i) how many drink tea and coffee both:

(ii) how many drink coffee but not tea.

(i) how many drink tea and coffee both:

(ii) how many drink coffee but not tea.

**Solution:**

Let A be the set of those persons who drink tea and let B be the set of those persons who drink coffee. Then,

A âˆ© B = set of persons who drink both tea and coffee.

A â€“ B = set of persons who drink tea but not coffee.

B â€“ A = set of persons who drink coffee but not tea

âˆ´n(A âˆª B) = 52, n(A-B) = 16 and n(A) = 33

Now, n(A-B) + n(A âˆ© B) = n(A)

**âˆ´**n(A âˆ© B) = n(A) - n(A â€“ B) = (33 â€“ 16) = 17

Thus, 17 persons drink tea and coffee both.

Now, n(A) = 33, n(A âˆª B) = 52 and n(A âˆ© B) = 17

âˆ´ n(A âˆª B) = n(A) + n(B) â€“ n(A âˆ© B)

**âˆ´**n(A) = 33, n (A âˆª B) = 52 and n(A âˆ© B) = 17.

**âˆ´**n(A âˆª B) = n(A) + n(B) â€“ n(A âˆ© B)

â‡’ n(B) = n(A âˆª B) + n(A âˆ© B) â€“ n(A)

â‡’ n(B) = (52 + 17 â€“ 33) =36

Also, n(B-A) + n(A âˆ© B) = n(B)

â‡’ n(B-A) = n(B) â€“ n(A âˆ© B) = (36 â€“ 17) = 19

**âˆ´**19 persons drink coffee but not tea.

# Question-45

**Represent the relation R from A = {2, 4, 5, 7} to B = {3, 5, 6, 8, 10} by an arrow diagram given a R b if b = a + 1 where a âˆˆ A and b âˆˆ B.**

**Solution:**

A = {2, 4, 5, 7} to B = {3, 5, 6, 8, 10}

# Question-46

**Let A = {1, 2, 3, 4}, B = {1, 2, 3} and C = {2, 4}. Find all sets X satisfying each pair of conditions:**

(i) X âŠ‚ B and X âŠ„ C

(i) X âŠ‚ B and X âŠ„ C

** (ii) X âŠ‚ A, X â‰ B and X âŠ„ C **

** (iii) X âŠ‚ A, X âŠ‚ B and X âŠ‚ C**

**Solution:**

(i) X = {1}, {3}, {1,2}, {1,3}, {2,3}, {1, 2, 3}

(ii) X = {1}, {3}, {1, 2}, {1, 3}, {2, 3}

(iii) Ï† , {2}

# Question-47

**Prove: (i) A â€“ B â‡” A âŠ† B and B âŠ† A.**

(ii) A âŠ† B and B âŠ† C â‡’ A âŠ† C

(ii) A âŠ† B and B âŠ† C â‡’ A âŠ† C

**Solution:**

(i) A â€“ B â‡” Every element of A is in B and every element of B is in A

â‡” A âŠ† B and B âŠ† A.

(ii) Let A âŠ† B and B âŠ† C.

And, let x be an arbitrary element of A.

Then x âˆˆ A â‡’ xâˆˆ B [Since A âŠ† B ]

â‡’ x âˆˆ C [Since B âŠ† C ]

âˆ´ A âŠ† C

# Question-48

**Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:**

(i) 1 âˆˆ A

(ii) {1, 2, 3} âŠ‚ A

(iii) {6, 7, 8} âˆˆA

(iv) {{4, 5}} âŠ‚ A

(v) Ï† âˆˆ A

(vi) Ï† âŠ‚ A

(i) 1 âˆˆ A

(ii) {1, 2, 3} âŠ‚ A

(iii) {6, 7, 8} âˆˆA

(iv) {{4, 5}} âŠ‚ A

(v) Ï† âˆˆ A

(vi) Ï† âŠ‚ A

**Solution:**

(i) False

(ii) False

(iii) True

(iv) True

(v) False

(vi) True

# Question-49

**If relation R is defined on N by a R b if b = 3a + 1, is R a finite set?**

**Solution:**

R = {(1, 4), (2, 7), (3, 10), (4, 13), â€¦â€¦â€¦â€¦}; R is a finite set.

# Question-50

**For any three sets A, B, C prove that: n (A âˆª B âˆª C) = n(A) + n(B) + n(C) + n(A âˆ© B âˆ© C) - [n(A âˆ© B)] â€“ [n(B âˆ© C)]â€“ [n(A âˆ© C)]**

**Solution:**

n (A âˆª B âˆª C) = n [(A âˆª B) âˆª C] = n(A âˆª B)+ n(C) - n [(A âˆª B) âˆ© C]

= n(A âˆª B)+ n(C) - n [(A âˆ© C) âˆª (B âˆ© C)]

= n (A)+ n ( B) â€“ n(A âˆ© B)+ n(C) - n [(A âˆ© C) + (B âˆ© C) â€“ n(A âˆ© C âˆ© B âˆ© C) ]

=** **n(A) + n(B) + n(C) + n(A âˆ© B âˆ© C) -[n(A âˆ© B)] â€“ [n(B âˆ© C)]â€“ [n(A âˆ© C)]

# Question-51

**How many elements has P(A) if A = Ï† ?**

**Solution:**

P(A) = {Ï† }. Therefore P(A) has 1 element.

# Question-52

**If x R y implies y = 15 â€“ 4x, given that x, y âˆˆ N, is R is finite set?**

**Solution:**

The set R = {(1, 11), (2, 7), (3, 3)}; R is a finite set.

# Question-53

**(ix) {{Ï† }} âŠ‚ A**

**Solution:**

(i) True

(ii) True

(iii) False, 1 is an element of A and not a set.

(iv) True

(v) False, 7 is an element of A and not a set.

(vi) True

(vii) True

(viii) True

(ix) True

# Question-54

**Examine what types of relations are the following (i) â€˜is of equal lengthâ€™ on the set of all line segments in a plane. (ii) â€˜is congruent toâ€™ on the set of triangles in a plane (iii) â€˜is a son ofâ€™ on the set of family members.**

**Solution:**

(i) â€˜is of equal lengthâ€™ on the set of all line segments in a plane is a reflexive, symmetric and transitive relation.

(ii) â€˜is congruent toâ€™ on the set of triangles in a plane is a reflexive, symmetric and transitive relation.

(iii) â€˜is a son ofâ€™ on the set of family members is not reflexive, not symmetric and not transitive relation.

# Question-55

**If X = {8**

^{n}â€“ 7n - 1; n âˆˆe} and Y = {49(n - 1);**n âˆˆN**, then prove that x âŠ† Y.**Solution:**

Let x

_{n}= 8

^{n }â€“ 7n - 1 = (1 + 7)

^{n}â€“ 7n - 1 =

for n â‰¤ 2

For x = 1, x_{n} = 0

Thus, X contains all positive integral multiplies of 49 of the form 40k_{n},

where k_{n} = .

Also, y contains all positive integral multiplies of 49 including zero.

Thus X âŠ† Y.

# Question-56

**Let A, B and C be three sets. If A âŠ‚ B and B âˆˆ C, is it true that A âˆˆ C? If not give an example.**

**Solution:**

No, A = {1, 2}, B = {1, 2, 3}, C = {{1, 2, 3}}

# Question-57

**Check which of the above relations is an equivalence relation.**

**Solution:**

(i) â€˜is of equal lengthâ€™ on the set of all line segments in a plane is an equivalence relation.

(ii) â€˜is congruent toâ€™ on the set of triangles in a plane is an equivalence relation.

(iii) â€˜is a son ofâ€™ on the set of family members is not an equivalence relation.

# Question-58

**For 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) A U B = {a,e,i,o,u,b,c}

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

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 âˆˆ N}

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

# Question-59

**Which of the following are reflexive? (a) â€˜is similar toâ€™ on the set of all triangles in a plane (b) â€˜is relatively prime toâ€™ on the set N (c) â€˜is parallel toâ€™ on the set of all lines on a plane (d) â€˜is less thanâ€™ on the set N.**

**Solution:**

(a) The relation â€˜is similar toâ€™ on the set of all triangles in a plane is reflexive.

(b) The relation â€˜is relatively prime to on the set N is not reflexive.

(c) The relation â€˜is parallel toâ€™ on the set of all lines on a plane is reflexive.

(d) The relation â€˜is less thanâ€™ on the set N not reflexive

# Question-60

**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-61

**Which of the following relations are symmetric? (a) â€˜is a sisterâ€™ on the set of all members of a family (b) â€˜is a multiple ofâ€™ on the set N (c) â€˜is a divisor of the set of all integers (d) â€˜is perpendicular toâ€™ on the set of all lines of a plane.**

**Solution:**

(a) The relation â€˜is a sisterâ€™ on the set of all members of a family is not symmetric.

(b) The relation â€˜is a multiple ofâ€™ on the set N is not symmetric.

(c) The relation â€˜is a divisorâ€™ on the set of all integers is not symmetric.

(d) The relation â€˜is perpendicular toâ€™ on the set of all lines on a plane is symmetric.

# Question-62

**If A = {2x/x âˆˆ N } and B = {2x+1 / x âˆˆ N} and subsets of the universal set X = N, find (i) Aâˆ© B (ii) Aâˆ© B (iii) Aâ€™ (iv) Bâ€™.**

**Solution:**

A = {2, 4, 6, 8â€¦â€¦}

B = {3, 5, 7, 9,â€¦.}

(i) Aâˆª B = {x/ x âˆˆ N, x â‰ 1}

(ii) Aâˆ© B = Ï•

(iii) Aâ€™ = {2x â€“ 1 / x âˆˆ N}

(iv) Bâ€™ = Aâˆ© {1}

# Question-63

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

**Solution:**

A U B = B.

# Question-64

**Which of the following relations are transitive? (a) â€˜is a friend ofâ€™ on the set of all students in a class (b) â€˜is congruent toâ€™ on the set of all triangles on a plane (c) â€˜is married toâ€™ on the set of all human beings (d) â€˜is relatively prime toâ€™ on the set N.**

**Solution:**

(a) The relation â€˜is a friend ofâ€™ on the set of all students in a class is not transitive.

(b) The relation â€˜is congruent toâ€™ on the set of all triangles on a plane is transitive.

(c) The relation â€˜is married toâ€™ on the set of all human beings is not transitive.

(d) The relation â€˜is relatively prime toâ€™ on the set N is not transitive.

# Question-65

**What type of relation R defined by (a) R = {(2, 2), (3, 2), (2, 3), (3, 3)} on A = {2, 3, 4} (b) R = {(- 1, 2), (2, - 1), (- 1, - 1), (2, 2), (3, 3)} on A = {- 1, 2, 3} (c) R = {(2, 3), (3, 3) (5, 5), (3, 2), (5, 7), (7, 5)} on A = {2, 3, 5, 7, 9}.**

**Solution:**

(a) The relation R = {(2, 2), (3, 2), (2, 3), (3, 3)} on A = {2, 3, 4} is symmetric and transitive.

(b) The relation R = {(- 1, 2), (2, - 1), (- 1, - 1), (2, 2), (3, 3)} on A = {- 1, 2, 3} is reflexive and transitive.

(c) The relation R = {(2, 3), (3, 3) (5, 5), (3, 2), (5, 7), (7, 5)} on A = {2, 3, 5, 7, 9} is only symmetric.

# Question-66

**For the following sets find their intersection :**

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

(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.}

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

(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.}

**Solution:**

i) A

ii) XY = {1,3}

iii) A = {3,6,9,12,â€¦â€¦â€¦}

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

.

^{.}. A B ={3}

# Question-67

**Prove that A - (A-B) = Aâˆ© B.**

**Solution:**

Let x âˆˆ A - (A-B)

â‡’x âˆˆ A and x âˆ‰ (A â€“ B)

â‡’x âˆˆ A and x âˆ‰ A and x âˆˆ B

â‡’x âˆˆ A and x âˆˆ B

â‡’x âˆˆ Aâˆ© B

âˆ´ A â€“ (A - B) âŠ‚ Aâˆ© B ------------------- (i)

Let x âˆˆ Aâˆ© B

â‡’ x âˆˆ A and x âˆˆ B

â‡’ x âˆˆ A and x âˆ‰ (A â€“ B)

â‡’ x âˆˆ A â€“ (A-B)

A âˆ© B âŠ‚ A â€“ (A-B) â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(ii)

From (i) and (ii)

A â€“ (A - B) = Aâˆ© B.

# Question-68

**Which of the following pairs of sets are 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:**

The pairs of sets in (iii) are disjoint.

# Question-69

**Check whether the following set defines an equivalence relation on the given set (a) R = {(p, p), (q, q), (r, r), (s, s), (p, q), (q, p), (s, r), (r, s)} on A = {p, q, r, s} (b) R = {(1, 3), (1, 1), (3, 1), (2, 3), (3, 3), (3, 2), (1, 2), (2, 1), (2, 2)} on A = {1, 2, 3}**

**Solution:**

(a) The relation R = {(p, p), (q, q), (r, r), (s, s), (p, q), (q, p), (s, r), (r, s)} on A = {p, q, r, s} is an equivalence relation.

(b) The relation R = {(1, 3), (1, 1), (3, 1), (2, 3), (3, 3), (3, 2), (1, 2), (2, 1), (2, 2)} on A = {1, 2, 3} is an equivalence relation.

# Question-70

**Given the set A = {5, 7, 9} write down a set of ordered pairs on A which defines an equivalence relation on A.**

**Solution:**

A = {5, 7, 9}

The set of all pairs of A = {(5, 5), (7, 7), (9, 9), (5, 7), (7, 5), (5, 9), (9, 5), (7, 9), (9, 7)}

# Question-71

**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

(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-72

**In the set N of natural numbers, examine the kinds of relation given below (i) If a**

^{2}+ b^{2}is a perfect square, a R b (ii) If a^{2}= b, a R b.**Solution:**

(i) If a

^{2}+ b

^{2}is a perfect square, a R b

(3, 4), (4, 3), (5, 12), (12, 5), (8, 6), (6, 8),

R is symmetric relation.

(ii) If a^{2} = b, a R b is not a relation.

# Question-73

**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-74

**(d) f = {(1, 2), (4, - 3), (9, 4), (16, 1)}**

**Solution:**

(a) A = {1, 4, 9, 16} to B = {- 1, 2, - 3, - 4}

f = {(1, - 1), (4, 2), (9, - 3), (16, - 4)} has one to one correspondence from the elements of A to B, f is a function from A to B.

(b) f = {(1, - 4) (1, - 1), (9, - 3), (16, 2)} does not have one to one correspondence from the elements of A to B, f is not a function from A to B.

(c) f = {(4, 2), (1, 2), (9, 2), (16, 2)} has one to one correspondence from the elements of A to B, f is a function from A to B.

(d) f = {(1, 2), (4, - 3), (9, 4), (16, - 1)} is a constant function, as every elements of A has same image in B, f is a function from A to B.

# Question-75

**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

(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-76

**X = {- 4, - 2, 0, 2, 4}, Y = {0, 1, 4, 9, 16} and f: X ###ERROR###Ã Y is defined by f(x) = x**

^{2}. Check whether f is a function.**Solution:**

f(x) = x

^{2 }

f(-4) = (- 4)^{2} = 16

f(-2) = (- 2)^{2} = 4

f(0) = (0)^{2} = 0

f(2) = (2)^{2} = 4

f(4) = (4)^{2} = 16

Since 16, 4, 0, 4 and 16 are the elements of Y, f: X ###ERROR###Ã Y is a function.

# Question-77

**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-78

**Is f: N ###ERROR###Ã N defined by f(x) = (x + 1) a function? Justify your answer.**

**Solution:**

f(x) = (x + 1)

f(1) = (1 + 1)

= (2)

= 1

f(2) = (2 + 1)

= (3)

= 1.5

When x = 2, f(2) = 1.5 is not an element of N since it is not a whole number.

âˆ´ f(x) = (x + 1) is not a function.

# Question-79

**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-80

**Find the range of the function f: A ###ERROR###Ã B where A = {- 3, - 2, - 1, 0, 1, 2}, B = {- 7 , - 4, - 3, - 1, 0, 2, 5, 8} given f(x) = 3x + 2.**

**Solution:**

f(x) = 3x + 2

f(- 3) = 3(- 3) + 2 = - 9 + 2 = - 7

f(- 2) = 3(- 2) + 2 = - 6 + 2 = - 4

f(- 1) = 3(- 1) + 2 = - 3 + 2 = - 1

f(0) = 3(0) + 2 = 0 + 2 = 2

f(1) = 3(1) + 2 = 3 + 2 = 5

f(2) = 3(2) + 2 = 6 + 2 = 8

âˆ´ The range of the function f: A ###ERROR###Ã B is {- 7, - 4, - 1, 2, 5, 8}

# Question-81

**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. **

**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-82

**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€™

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-83

**Find the images of â€“ 1, - 2, 3, 4 of the function f: Z ###ERROR###Ã Z is defined by f(x) = 3x.**

**Solution:**

f(x) = 3x

f(- 1) = 3(- 1) = - 3

f(- 2) = 3(- 2) = - 6

f(3) = 3(3) = 9

f(4) = 3(4) = 12

The images of the function f(x) = 3x are â€“3, - 6, 9, 12.

# Question-84

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

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

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

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

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

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

**Solution:**

(i) Aâ€™ = {d, e, f, g, h}

(ii) Bâ€™ = {a, b, c, h}

(iii) Câ€™ = {b, d, f, h}

(iv) Dâ€™ = {b, c, d, e}

# Question-85

**If f: N ###ERROR###Ã N is defined by f(x) = x**

^{2}- 1 check whether f is a function. If f is not a function, find for what choice of the codomain, f will be a function.**Solution:**

f(x) = x

^{2}â€“ 1

f(1) = (1)^{2} â€“ 1 = 0

âˆ´ f(x) = x^{2} â€“ 1 is not a function since 0 is not an element of N.

If the codomain is the set of whole numbers, then f will be a function.

# Question-86

**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}

(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 âˆˆ N 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-87

**A = {0, 1, 2, 3, 4,}, B = {- 6, - 3, 0, 6}. Relation f: A ###ERROR###Ã B is such that f(x) = - 3x. Check whether it is a function.**

**Solution:**

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

f(x) = - 3x

f (0) = - 3 (0) = 0

f(1) = - 3(1) = - 3

f(2) = - 3(2) = - 6

f(3) = - 3(3) = - 9

f(4) = - 3(4) = - 12

The elements of B are â€“6, -3, 0 and 6 f: A ###ERROR###Ã B is such that f(x) = - 3x is not a function.

# Question-88

**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€™

(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-89

**{(3, 10), (- 3, 4), (1, 8), (0, 7), (- 5, 2)}. Is this relation set a function?**

**Solution:**

A = {3, - 3, 1, 0, - 5} and B = {10, 4, 8, 7, 2}

This relation is a function.

# Question-90

**Show that (A âˆª B) â€“ (A âˆ© B) = (A - B) âˆª (B - A).**

**Solution:**

L.H.S = (A âˆª B) â€“ (A âˆ© B) = (A âˆª B) âˆ© (A âˆ© B)â€™ = (A âˆª B) âˆ© (Aâ€™ âˆª Bâ€™) = (A âˆª B âˆ© A') âˆª (A âˆ© B âˆª Bâ€™)

= (B âˆ© Aâ€™) âˆª(A âˆ© Bâ€™)

R.H.S = (A - B) âˆª (B - A) = (A âˆ© Bâ€™) âˆª (B âˆ© Aâ€™) = (A âˆ© Bâ€™) âˆª (B âˆ© Aâ€™)

Therefore L.H.S = R.H.S

# Question-91

**In the function f(x) = x**

^{2}â€“ x + 7, the domain of f is {1, 3, - 3} find the range of f.**Solution:**

The domain of f is {1, 3, - 3}

f(x) = x^{2} â€“ x + 7

f(1) = (1)^{2} â€“ 1 + 7 = 1 â€“ 1 + 7 = 7

f(3) = (3)^{2} â€“ 3 + 7 = 9 â€“ 3 + 7 = 13

f(- 3) = (- 3)^{2} â€“ (- 3) + 7 = 9 + 3 + 7 = 19

âˆ´ The range of the function = {7, 13, 19}

# Question-92

**Find the images of X = {125, 64, 1} if f: X ###ERROR###Ã Y is defined by f(x) = cube root of x where Y = {1, 2, 3, 4, 5}.**

**Solution:**

X = {125, 64, 1}

Cube root of 125 = 5

Cube root of 64 = 4

Cube root of 1 = 1

âˆ´ The images are 5, 4, 1.

# Question-93

**Shade the following sets:**

Aâ€™ âˆ© (B âˆª C)

**Solution:**

# Question-94

**Shade the following sets:**

Aâ€™ âˆ© (C - B) in the Venn diagram given in figure below

**Solution:**

# Question-95

**Find the pre-image of 2 under the function f = {0, - 1), (3, 2), (5, 3), (7, 2)}.**

**Solution:**

f = {(0, - 1), (3, 2), (5, 3), (7, 2)}

(3, 2) and (7, 2) are having the element 2.

âˆ´ The pre-images of 2 are 3 and 7.

# Question-96

**Is the function from N ###ERROR###Ã N given by f(x) = 2x + 1, onto? Write down the range.**

**Solution:**

f(x) = 2x + 1

f(1) = 2(1) + 1 = 3

f(2) = 2(2) + 1 = 5

f(3) = 2(3) + 1 = 7

âˆ´ The range of the function f(x) = 2x + 1 is {3, 5, 7, â€¦â€¦â€¦}.

# Question-97

**If X = {4, 6, 8, 10}, Y = {3, 4, 5, 6, 7} and f: X ###ERROR###Ã Y is given by f(x) = + 1 represent f as (i) a set of ordered pairs (ii) a table.**

**Solution:**

X = {4, 6, 8, 10}, Y = {3, 4, 5, 6, 7}

(i) f(x) = + 1

f(4) = + 1 = 3

f(6) = + 1 = 4

f(8) = + 1 = 5

f(10) = + 1 = 6

âˆ´ The ordered pairs of the function f: {(4, 3), (6, 4), (8, 5), (10, 6)}

(ii)

x |
4 |
6 |
8 |
10 |

y |
3 |
4 |
5 |
6 |

# Question-98

**If A = {0, 1, 2, 3}, B = {3, 7, 11, 15, 17}, f: A ###ERROR###Ã B is defined by f(x) =4x + 3, represent f as (i) the set of ordered pairs (ii) a table (iii) a graph (iv) an arrow diagram.**

**Solution:**

(i) A = {0, 1, 2, 3}, B = {3, 7, 11, 15, 17}

The set of ordered pairs = {(0, 3), (1, 7), (2, 11), (3, 15)}

(ii)

x |
0 |
1 |
2 |
3 |

y |
3 |
7 |
11 |
15 |

(iii)

(iv)

# Question-99

**(a) write its domain and range. Represent it using (b) an arrow diagram (c) a table.**

**Solution:**

(a) Domain = {- 1, - 3, - 5, - 4}

Range = {2, 1, 6, 3}

(b)

(c)

x |
- 1 |
- 3 |
- 5 |
- 4 |

y |
2 |
1 |
6 |
3 |

# Question-100

**Decide, among the following sets, which are the subsets of which**

** A = {all real numbers satisfying x ^{2} - 8x + 12 = 0} **

** B = {2, 4, 6} **

** C = {2, 4, 6, 8, ...} **

** D = {6}**

**Solution:**

A = {all real numbers satisfying x

^{2}- 8x + 12 = 0} = {2, 6}

B = {2, 4, 6}

C = {2, 4, 6, 8, ...}

D = {6}.

A âŠ‚ B; A âŠ‚ C; B âŠ‚ C; D âŠ‚ A; D âŠ‚ B; D âŠ‚ C.

# Question-101

**In each of the following, determine whether the statement is true or false . If it is false, give an example.**

(i) If x âˆˆ A and A âˆˆ B, then x âˆˆ B.

(ii) If A âŠ‚ B and B âˆˆ C, then A âˆˆ C.

(i) If x âˆˆ A and A âˆˆ B, then x âˆˆ B.

(ii) If A âŠ‚ B and B âˆˆ C, then A âˆˆ C.

** (iii) If A âŠ‚ B and B âŠ‚ C, then A âŠ‚ C
(iv) If A âŠ„ B and B âŠ„ C, then A âŠ„ C. **

** (v) If x âˆˆ A and A âŠ„ B, then x âˆˆ B.
(vi) If A âŠ‚ B and x âˆ‰ B, then x âˆ‰ A. **

**Solution:**

(i) False

Let A = {x, 1}, B = {{x, 1}, 2}

Then x âˆ‰ B.

(ii) False

Let A = {1, 2}, B = {1, 2, 3}, C = {{1, 2, 3}, 5}

Then {1, 2} âˆ‰ C

(iii) True

(iv) False

Let A = {1, 2}, B = {3, 4}, C = {1,2,3}

Then A âŠ‚C

(v) False

Let A = {x, 1}, B = {3, 4}

Then x âˆ‰ B

(vi) True

# Question-102

**Let B be a subset of A and let P(A:B) = {Xâˆˆ P(A)| XâŠƒ B }.**

(i) Let B = {a, b} and A = {a, b, c, d}. List all the members of the set P(A:B).

(i) Let B = {a, b} and A = {a, b, c, d}. List all the members of the set P(A:B).

** (ii) Show that P(A: Ï† ) = P(A). **

**Solution:**

(i) {a, b}, {a, b, c}, {a, b, d}, {a, b, c, d}

(ii) P(A: Ï† ) ={{a},{b},{c},{d},{a, b}, {b,c},{c,d},{a,d},{b,d},{a, b, c}, {a, b, d},{a, b, c, d}}

P(A) ={{a},{b},{c},{d},{a, b}, {b,c},{c,d},{a,d},{b,d},{a, b, c}, {a, b, d}, {a, b, c, d}}

âˆ´ P(A: Ï† ) = P(A)

# Question-103

**Show that for any sets A and B, A = (A âˆ© B) âˆª (A - B) and A âˆª (B - A) = A âˆª B.**

**Solution:**

A = A âˆ© U

= A âˆ© (B âˆª Bâ€™)

= (A âˆ© B) âˆª (A âˆ© Bâ€™)

= (A âˆ© B) âˆª (A - B)

A âˆª (B - A) = A âˆª (B âˆ© Aâ€™)

= (A âˆª B) âˆ© (A âˆª Aâ€™)

= (A âˆª B) âˆ© U

= A âˆª B

# Question-104

**Using properties of sets, prove that A âˆª (B âˆ© A) = A**

**Solution:**

A âˆª (B âˆ© A) = (A âˆª B) âˆ© (A âˆª A) = U âˆ© A = A

# Question-105

**Using properties of sets, prove that A âˆ© (A âˆª B) = A**

**Solution:**

A âˆ© (A âˆª B) = (A âˆ© A) âˆª (A âˆ© B) = A âˆª U = A