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#### Question 1:

Write the following sets in roster form.
(i) Set of even numbers
(ii) Set of even prime numbers from 1 to 50
(iii) Set of negative integers
(iv) Seven basic sounds of a sargam (sur)

(i) A = {..., $-$6, $-$4, 0, 2, 4, 6, ...}

(ii) B = {2}

(iii) C = {..., $-$5, $-$4, $-$3, $-$2, $-$1}

(iv) D = {SA, RE, GA, MA, PA, DHA, NI}

#### Question 2:

Write the following symbolic statements in words.
(i) $\frac{4}{3}\in$ Q    (ii) $-2\notin$N  (iii) P = {p  |  p is an odd number}

(i) $\frac{4}{3}\in$ Q

$\frac{4}{3}$ belongs to set Q.

(ii) $-2\notin$ N

$-2$ does not belongs to set N.

(iii) P = {p  |  p is an odd number}

P is a set of p such that p is an odd number.

#### Question 3:

Write any two sets by listing method and by rule method.

(i) A = {6, 7, 8, 9}

A = {xx $\in$ N and 5 < x < 10}

(ii) B = {a, e, i, o, u}

B = {y | y is vowel of English alphabet}

#### Question 4:

Write the following sets using listing method.

(i) All months in the indian solar year.
(ii) Letters in the word ‘COMPLEMENT’.
(iii) Set of human sensory organs.
(iv) Set of prime numbers from 1 to 20.
(v) Names of continents of the world.

(i) A = {January, February, March, April, May, June, July, August, September, October, November, December}

(ii) B = {C, O, M, P, L, E, M, E, N, T}

(iii) C = {Nose, Tongue, Skin, Ear, Eyes}

(iv) D = {2, 3, 5, 7, 11, 13, 17, 19}

(v) E = {Asia, Africa, Europe, North America, South America, Australia, Antartica}

#### Question 5:

Write the following sets using rule method.
(i) A = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
(ii) B = { 6, 12, 18, 24, 30, 36, 42, 48}
(iii) C = {S, M, I, L, E}
(iv) D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
(v) X = {a, e, t}

(i) A = {x | x = n2n $\in$ N and $1\le n\le 10$}

(ii) B = {x | x = 6yy $\in$ N and $1\le y\le 8$}

(iii)  C = Set of letters in the word 'SMILE'

(iv)  D = Set of days in a week

(v)   X = Set of letters in the word 'ate'

#### Question 1:

Decide which of the following are equal sets and which are not ? Justify your answer.
A = { x   | 3 x -1 = 2}
B = { x   |  x is a natural number but x is neither prime nor composite}
C = { x   |  x $\in$ N, x < 2}

Since, A = {x | 3x $-$ 1 = 2} = {1};

B = {x | x is a natural number but x is neither prime nor composite} = {1}; and

C = {x | x $\in$ N, x < 2} = {1}

So, A = B = C

#### Question 2:

Decide whether set A and B are equal sets. Give reason for your answer.

A = Even prime numbers
B = { x  |  7 x -1 = 13}

Since, A = Even prime numbers = {2}; and

B = {x | 7x $-$ 1 = 13} = {2}

So, A = B

hence, A and B are equal sets.

#### Question 3:

Which of the following are empty sets ? why ?
( i ) A = { a  |  a is a natural number smaller than zero.}
( ii ) B = { x | x 2 = 0}
( iii ) C = { x | 5 x - 2 = 0, x$\in$N}

(i) A = {a | a is a natural number smaller than zero.} = {}

So, A is an empty set.

(ii) B = {x | x2 = 0} = {0}

So, B is not an empty set.

(iii) C = {x | 5x $-$ 2 = 0, x$\in$N} = {}

So, C is an empty set.

#### Question 4:

Write with reasons, which of the following sets are finite or infinite.
( i ) A = { x | x < 10, x is a natural number}
(ii) B = { y | y < -1, y is an integer}
(iii) C = Set of students of class 9 from your school.
(iv) Set of people from your village.
(v) Set of apparatus in laboratory
(vi) Set of whole numbers
(vii) Set of rational number

(i) A = {x | x < 10, x is a natural number} = {1, 2, 3, 4, 5, 6, 7, 8, 9}

So, A is a finite set.

(ii) B = {y | y < $-$1, y is an integer} = {..., $-$5, $-$4, $-$3, $-$2}

So, B is an infinite set.

(iii) C = Set of students of class 9 from your school.

Since, the number of elements of set C is countable number.

So, C is a finite set.

(iv) D = Set of people from your village.

Since, the number of elements of set D is countable number.

So, D is a finite set.

(v) E = Set of apparatus in laboratory

Since, the number of elements of set E is countable number.

So, E is a finite set.

(vi) F = Set of whole numbers = {0,1, 2, 3, 4, ...}

So, F is an infinite set.

(vii) G = Set of rational number

Since, the number of elements of set G is infinite.

So, G is an infinite set.

#### Question 1:

If A = { a , b, c, d, e}, B = { c, d, e, f  }, C = {b, d}, D = { a , e}

then which of the following statements are true and which are false ?
( i ) C $\subseteq$ B ( ii ) A $\subseteq$ D ( iii ) D $\subseteq$B ( iv ) D$\subseteq$A ( v ) B $\subseteq$A (vi) C $\subseteq$A

We have, A = {a , b, c, d, e}, B = {c, d, e, f }, C = {b, d}, D = {a, e}

(i) Since,

So, C $\subseteq$ B is false.

(ii) Since,

So, A $\subseteq$ D is false.

(iii) Since,

So, D $\subseteq$ B is false.

(iv) Since, all elements of set D is in set A.

So, D $\subseteq$ A is true.

(v) Since,

So, B $\subseteq$ A is false.

(vi) Since, all elements of set C is in set A.

So, C $\subseteq$ A

#### Question 2:

Take the set of natural numbers from 1 to 20 as universal set and show set X and Y using Venn diagram.

( i ) X = { x   |   x$\in$N, and 7 < x < 15}
( ii ) Y = { y   |   y $\in$N, y is prime number from 1 to 20}

We have,

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20};

X = {x | x$\in$ N, and 7 < x < 15} = {8, 9 ,10, 11, 12, 13, 14}; and

Y = {y | y $\in$N, y is prime number from 1 to 20} = {2, 3, 5, 7, 11, 13, 17, 19} #### Question 3:

U = {1, 2, 3, 7, 8, 9, 10, 11, 12}
P = {1, 3, 7, 10}
then
(i) show the sets U, P and P ¢ by Venn diagram.
(ii) Verify (P ¢ ) ¢ = P

We have,

U = {1, 2, 3, 7, 8, 9, 10, 11, 12} and P = {1, 3, 7, 10}

P' = U $-$ P = {2, 4, 8, 9, 11, 12}

(i) (ii) (P')' = U $-$ P' = {1, 3, 7, 10} = P

#### Question 4:

A = {1, 3, 2, 7} then write any three subsets of A.

We have,

A = {1, 3, 2, 7}

The three subsets of A are: {1}, {2} and {1, 2, 3}.

Disclaimer: There are 24 = 16 subsets possible i.e. {}, {1}, {2}, {3}, {7}, {1, 2}, {1, 3}, {1, 7}, {2, 3}, {2, 7}, {3, 7}, {1, 2, 3}, {2, 3, 7}, {1, 3, 7}, {1, 2, 7} and {1, 2, 3, 7}.

#### Question 5:

( i)   Write the subset relation between the sets.
P is the set of all residents in Pune.
I is the set of all residents in Indore.
B is the set of all residents in India.
H is the set of all residents in Maharashtra.

( ii)  Which set can be the universal set for above sets ?

(i) Sine Pune is a city in Maharashtra, and Maharashtra is a state in India.

So, P $\subset$ H $\subset$ B

Similarly,

So, I $\subset$ M $\subset$ B

(ii) The set B can be the universal set for above sets.

#### Question 6:

Which set of numbers could be the universal set for the sets given below?

(i) A = set of multiples of 5, B = set of multiples of 7.
C = set of multiples of 12

(ii) P = set of integers which are multiples of 4. T = set of all even square numbers.1

(i) We have,

A = set of multiples of 5 = {5, 10, 15, ...};  B = set of multiples of 7 = {7, 14, 21, ...}; and C = set of multiples of 12 = {12, 24, 36, ...}

The universal set for the sets A, B and C can be set of natural numbers, i.e. N.

Disclaimer: The Universal set for the sets A, B and C can be set of whole numbers, integers, rational number of real numbers and so on.

(ii) We have,

P = set of integers which are multiples of 4 = {4, 8, 12, ...}; and T = set of all even square numbers = {4, 16, 36, 64, 100, ...}

So, P can be the Universal set for the sets P and Q.

Disclaimer: Here also, the Universal set can be vary from the ablove.

#### Question 7:

Let all the students of a class is an Universal set. Let set A be the students who secure 50% or more marks in Maths. Then write the  complement of set A.1

We have,

U = set of all the students of a class and A = set of the students who secure 50% or more marks in Maths

So, the complement of set A, A' = U $-$ A = set of the students who secure less than 50% marks in Maths.

#### Question 1:

If n (A) = 15, n (A$\cup$B ) = 29, n (A $\cap$ B) = 7 then n (B) = ?

We have,

n (A) = 15, n (A$\cup$B ) = 29, n (A $\cap$ B) = 7

Since, n (B) = n (A$\cup$B ) + n (A $\cap$ B) $-$ n (A) =

So, n (B) = 21

#### Question 2:

In a hostel there are 125 students, out of which 80 drink tea, 60 drink coffee and 20 drink tea and coffee both. Find the number of students who do not drink tea or coffee.

Let A be the set of students who drink tea, and B be the set of setudents who drink coffee.

We have,

n(U) = 125, n(A) = 80, n(B) = 60,

Since,

So, the number of students who do not drink tea or coffee = 125 $-$ 120 = 5

#### Question 3:

In a competitive exam 50 students passed in English. 60 students passed in Mathematics. 40 students passed in both the subjects. None of them fail in both the subjects. Find the number of students who passed at least in one of the subjects ?

Let A be the set of students who passed in English, and B be the set of students who passed in Mathematics.

We have,

n(A) = 50, n(B) = 60,

Since,

So, the number of students who passed at least in one of the subjects is 70.

#### Question 4:

A survey was conducted to know the hobby of 220 students of class IX. Out of which 130 students informed about their hobby as rock climbing and 180 students informed about their hobby as sky watching. There are 110 students who follow both the hobbies. Then how many students do not have any of the two hobbies ? How many of them follow the hobby of rock climbing only ? How many students follow the hobby of sky watching only ?

Let A be the set of students who follow the hobby of rock climbing, and B be the set of students who follow the hobby of sky watching.

We have, n (A) = 130, n (B) = 180, n (A $\cap$ B) = 110, n (U) = 220

Since,

n (A $\cup$ B) = n (A) + n (B) $-$ n (A $\cap$ B) = 130 + 180 $-$ 110 = 200

So, the number of students who do not have any of the two hobbies = n (U) $-$ n (A $\cup$ B) = 220 $-$ 200 = 20

Also, the number of students who follow the hobby of rock climbing only = n (A) $-$ n (A $\cap$ B) = 130 $-$ 110 = 20

And, the number of students who follow the hobby of sky watching only = n (B) $-$ n (A $\cap$ B) = 180 $-$ 110 = 50

#### Question 5:

Observe the given Venn diagram and write the following sets. (i) A    (ii) B   (iii) A$\cup$B (iv) U

(v) A'    (vi) B' (vii) (A$\cup$B) '

We have, (i) A = {m, n, x, y, z}

(ii) B = {m, n, p, q, r}

(iii) A$\cup$= {m, n, x, y, z, p, q, r}

(iv) U = {m, n, x, s, t, y, z, p, q, r}

(v) A' = {m, n, s, t, p, q, r}

(vi) B' = {m, n, x, s, t, y, z\}

(vii) (A$\cup$B)' = {s, t}

#### Question 1:

Choose the correct alternative answer for each of the following questions.

(i) If M = {1, 3, 5}, N = {2, 4, 6}, then M$\cap$N = ?

(A) {1, 2, 3, 4, 5, 6}  (B) {1, 3, 5}  (C)  $\varphi$     (D) {2, 4, 6}

(ii) P = { x | x is an odd natural number, 1 < x $\le$5} How to write this set in roster form?

(A) {1, 3, 5}  (B) {1, 2, 3, 4, 5} (C) {1, 3}  (D) {3, 5}

(iii) P = {1, 2, ........., 10}, What type of set P is ?

(A) Null set  (B) Infinite set  (C) Finite set  (D) None of these

(iv) M$\cup$N = {1, 2, 3, 4, 5, 6} and M = {1, 2, 4} then which of the following represent set N ?

(A) {1, 2, 3} (B) {3, 4, 5, 6} (C) {2, 5, 6}

(D) {4, 5, 6}

(v) If P$\subseteq$ M, then Which of the following set represent P $\cap$ (P $\cup$ M) ?

(A) P   (B) M   (C) P$\cup$M  (D) P' $\cap$ M

(vi) Which of the following sets are empty sets ?
(A) set of intersecting points of parallel lines
(B) set of even prime numbers.
(C) Month of an english calendar having less than 30 days.
(D) P = { x  |   x $\in$ I, $-$1 < <  1}

(i) We have,

M = {1, 3, 5}, N = {2, 4, 6}

M$\cap$N = $\varphi$ = Empty set

So, the correct option is (C).

(ii) Since, P = {x | x is an odd natural number, 1 < x $\le$5} = {3, 5}

So, the correct option is (D).

(iii) Since, the elements of set P = {1, 2, ..., 10} is finite.

So, set P is a finite set.

Hence, the correct option is (C).

(iv) We have, $\cup$ N = {1, 2, 3, 4, 5, 6} and M = {1, 2, 4}

Since, {1, 2, 3} $\cup$ {1, 2, 4} = {1, 2, 3, 4} $\ne$ M $\cup$ N = {1, 2, 3, 4, 5, 6};

{3, 4, 5, 6} $\cup$ {1, 2, 4} = {1, 2, 3, 4, 5, 6} = M $\cup$ N = {1, 2, 3, 4, 5, 6};

{2, 5, 6} $\cup$ {1, 2, 4} = {1, 2, 4, 5, 6} $\ne$ M $\cup$ N = {1, 2, 3, 4, 5, 6}; and

{4, 5, 6} $\cup$ {1, 2, 4} = {1, 2, 4, 5, 6} $\ne$ M $\cup$ N = {1, 2, 3, 4, 5, 6}

So, the correct option is (B).

(v) We have, P$\subseteq$ M,

Now, P $\cap$ (P $\cup$ M) = P $\cap$ M = M         (Since, P $\cup$ M = M; P$\subseteq$ M)

So, the correct option is (B).

(vi) Since,

the set of intersecting points of parallel lines = {};

the set of even prime numbers= {2};

the Month of an english calendar having less than 30 days = {February}; and

P = {x | x $\in$ I, $-$1 <  x <  1} = {0}

So, the correct option is (A).

#### Question 2:

Find the correct option for the given question.

(i) Which of the following collections is a set ?
(A) Colours of the rainbow (B) Tall trees in the school campus. (C) Rich people in the village (D) Easy examples in the book

(ii) Which of the following set represent N $\cap$ W?
(A) {1, 2, 3, .....}   (B) {0, 1, 2, 3, ....}   (C) {0}    (D) { }

(iii) P = {  |   x is a letter of the word ' indian'} then which one of the following is set P in listing form ?
(A) {i, n, d}  (B) {i, n, d, a} (C) {i, n, d, i, a} (D) {n, d, a}

(iv) If T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8} then T$\cup$M = ?
(A) {1, 2, 3, 4, 5, 7} (B) {1, 2, 3, 7, 8}
(C) {1, 2, 3, 4, 5, 7, 8} (D) {3, 4}

(i)

(A) Since, the colours of the rainbow are well defined such as Violet, Indigo, Blue, Green, Yellow, Orange and Red.

So, the collection of colours of the rainbow is a set.

(B) Since, the tall trees in the school campus are not well defined.

So, the collection of the tall trees in the school campus is not a set.

(C) Since, the rich people in the village is not well defined.

So, the colection of the rich people in the village is not a set.

(D) Since, the easy examples in the book is not well defined.

So, the collection of easy examples in the book is not a set.

(ii) Since, N $\cap$ W = {1, 2, 3, 4, ...}

So, the correct option is (A).

(iii) Since, P = { x is a letter of the word 'indian'}

So, P = {i, n, d, i, a, n} = {i, n, d, a}

Hence, the correct option is (B).

(iv) Since, T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8}

So, T$\cup$M = {1, 2, 3, 4, 5, 7, 8}

Hence, the correct option is (C).

#### Question 3:

Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speak at least one language. Then how many speak only English ? How many speak only French ? How many of them speak English and French both ?

Let A be the set of persons speaking English and B be the set of persons speaking French.

So, n (A) = 72; n (B) = 43;

Now,

Also,

And,

#### Question 4:

70 trees were planted by Parth and 90 trees were planted by Pradnya on the occasion of Tree Plantation Week. Out of these; 25 trees were planted by both of them together. How many trees were planted by Parth or Pradnya ?

Let A be the set of tress planted by Parth and B be the set of trees planted by Pradnya.

So, n (A) = 70; n (B) = 90;

Now,

݅݅݅

Hence, the number of trees planted by Parth or Pradnya is 135.

#### Question 5:

If n (A) = 20, n (B) = 28 and n (A$\cup$B) = 36 then n (A $\cap$ B) = ?

We have,

n (A) = 20, n (B) = 28 and n (A$\cup$B) = 36

Since, n (A $\cap$ B ) = n (A) + n (B) $-$ n (A $\cup$ B) =

$\therefore$ n (A $\cap$ B ) = 12

#### Question 6:

In a class, 8 students out of 28 have a dog as their pet animal at home, 6 students have a cat as their pet animal. 10 students have dog and cat both, then how many students do not have a dog or cat as their pet animal at home ?

We have,

Total number of  students = 28;

Students have a dog as their pet = 8;

Students have a cat as their pet = 6; and

Students have cat and dog both = 10

Solving using venn diagram, we get: So, the number of students that do not have a dog or a cat as their pet is 4.

#### Question 7:

Represent the union of two sets by Venn diagram for each of the following.

(i) A ={3, 4, 5, 7} B ={1, 4, 8}
(ii) P = {a, b, c, e, f} Q ={l, m, n , e, b}
(iii) X = { x | x is a prime number between 80 and 100}
Y = { y | y is an odd number between 90 and 100 }

(i) A ={3, 4, 5, 7}  B ={1, 4, 8} (ii) P = {ab, c, ef}  Q ={lmn , eb} (iii)  X = { x | x is a prime number between 80 and 100} = {83, 89, 97};
Y = { y y is an odd number between 90 and 100 } = {91, 93, 95, 97, 99} #### Question 8:

Write the subset relations between the following sets..

X = set of all quadrilaterals. Y = set of all rhombuses.
S = set of all squares. T = set of all parallelograms.
V = set of all rectangles.

Since, all squares are rectangle, all rectangles are parallelogram, all parallelograms are quadrilateral; and all squares are rhombus, all rhombus are parallelogram, all parallelograms are quadrilateral.

So, the subset relations are:

S < V < T < X and S < Y < T < X

#### Question 9:

If M is any set, then write M  and M .

If M is any set, then

and,

#### Question 10: Observe the Venn diagram and write the given sets
U, A, B, A $\cup$ B, A $\cap$ B

(i) U = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13}

(ii) A = {1, 2, 3, 5, 7}

(iii) B = {1, 5, 8, 9, 10}

(iv) A $\cup$ B = {1, 2, 3, 5, 7, 8, 9, 10}

(v)

#### Question 11:

If n (A) = 7, n (B) = 13, n (A $\cap$ B )= 4, then n (A $\cup$ B)=?

n (A) = 7, n (B) = 13 and n (A $\cap$ B) = 4
Since, n (A $\cup$ B) = n (A) + n (B) $-$ n (A $\cap$ B) = 7 + 13 $-$ 4
$\therefore$ n (A $\cup$ B) = 16