Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-1 (a),  Exe-1 (b), Exe-1 (c), Exe-1 (d), and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1.

Class: 11th
Subject: Mathematics
Chapter  : Ch-1 Sets of Section -A
Board ISC
Writer  OP Malhotra
Publications S.Chand Publications 2020-21

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Exe-1 (a),

 Exe-1 (b),

 Exe-1 (c),

 Exe-1 (d),

Chapter Test


What is Set

A set is a group or collection of objects or numbers, considered as an entity unto itself. Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z. Each object or number in a set is called a member or element of the set. Examples include the set of all computers in the world, the set of all apples on a tree, and the set of all irrational numbers between 0 and 1

Important Equations

  1. For any two sets P and Q,
  • (P ∪ Q)′ = P′ ∩ Q′
  • (P ∩ Q)′ = P′ ∪ Q′
  1. If P and Q are finite sets such that P ∩ Q = φ, then n (P ∪ Q) = n (P) + n (Q).
  2. If P ∩ Q ≠ φ, then

n (P ∪ Q) = n (P) + n (Q) – n (P ∩ Q)

  1. n (P ∪ Q ∪ R) = n(P) + n(Q) + n(R) – n(P ∩ Q) – n(P ∩ Q) – n(P ∩ Q ) + n(P ∩ Q ∩ R)
  2. If P is a subset of set U (Universal Set), then its complement (P′) is also a subset of Universal Set (U).

Some Properties of Operation of Intersection

  • P ∩ Q = Q ∩ P (Commutative law).
  • (P ∩ Q) ∩ R = P ∩ (Q ∩ R) (Associative law).
  • φ ∩ P = φ, U ∩ P = P.
  • P ∩ P = P (Idempotent law).
  • P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ Q) (Distributive law).

Some Properties of the Operation of Union

    • P ∪ Q = Q ∪ P (Commutative law).
    • (P ∪ Q) ∪ R = P ∪ ( Q ∪ R) (Associative law).
    • P ∪ φ = P (Law of the identity element).
    • U ∪ P = U (Law of U).

Exe-1 (a),

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1.

Page-Number 1-

Question-1

Which of the collections are sets ?
(i) The collection of all months of a year, beginning with letter J
(ii) The collection of most talented writers of India.
(iii) The collection of all natural numbers less than 100.
(iv) A collection of most dangerous animals of the world.

Answer-1

(i)  it is a well defined collection of objects so it represents a set.
(ii)  it is not a well defined collection of objects as the word the most talented is vague. As according  to some people, those writers are not talented. Hence given collection does not represents a set.
(iii) The collection of all natural numbers less than 100 = {1, 2, 3, …, 99} clearly it is a well defined collection so it represents a set.
(iv) people may have different opinions. Hence given collection does not represents a set.

Question-2

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol  and in the blank spaces.
(i) 5 …. A, (ii) 8 …. A, (iii) 0 …. A (iv) 4 …. A

Answer-2

Given A = {1, 2, 3, 4, 5, 6}
(i) Since 5 is a member of set A, Hence 5 ∈ A
(ii) 8 is not a member of set A, hence 8 ∉ A
(iii) 0 is not an element of set, A hence 8 ∉ A
(iv) 4 is an element of set A Hence 4 ∈ A

Question-3

. Write down a description of each of the following sets. (There could be different suitable descriptions.)
(i) {2, 4, 6, 8}
(ii) {7, 14, 21, 28, 35}
(iii) {1, 2, 3, 4, 6, 12}

Answer-3

(i) {2, 4, 6, 8} = set of all even numbers from 2 to 8
(ii) {7, 14, 21, 28, 35} = set of all multiples of 7 from 7 to 35.
(iii) {1,2,3,4,6,12}= set of all factors of 12

Question-4

.List the following sets in roster form.
(i) The set of square numbers less than 40.
(ii) The set of colours of the rainbow.
(iii)

    (a) The set of factors of 144.
(b) The set of prime factors of 144.
(iv) The set of natural numbers less than 50.
(v) The set of consonants before i in the English alphabet.

(vi) The set of letters in the word ‘Satellite’.

Answer-4

(i) {1, 4, 9, 6, 16, 25, 36}
(ii) Indigo, Violet, Brown, Green, Yellow, Orange, Red}
(iii)

(a) 144 = factor are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 72, 144
So Required set in Roster form
= {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 72, 144}
(b) {2, 3}
(iv) {1, 2, 3, 4, 5, ……, 49}
(v) {b, c, d, f, g, h}
(vi) {s, a, t, e, l, i}

Question-5

. Rewrite the following sets in the indicated notation.
(i) {– 2, – 4, – 6, – 8} ; (a) Words (b) Set-builder notation
(ii) Positive multiple of 11 ; Roster form
(iii) {– 9, – 7, – 5, – 3, – 11} ; Set-builder notation
(iv) Even numbers between 27 and 39 (a) Roster form (b) Set-builder notation
(v) {x | 0 < x < 1} ; (a) Words (b) Roster form
(vi) –5 –4 –3 –2 –1; Roster form
(vii) Negative multiples of 3, Set-builder form
(viii) –4 –3 –2 –1 ; Set-builder form
(ix) Numbers more than 2 units from 8, Set-builder form.
(x) x ¹ 5 and x £ 10 ; Set-builder form
(xi) {y | y = 5x – 2, y ∈ N} ; Roster form
(xii)……………………Roster form

Answer-5

(i)

 (a) set of all even numbers between – 9 and 0.
(b) {x : x < 0 and x is even}

(ii) {11, 22, 33, 44, ……}
(iii) {x : – 9 ..≤ x.≤ .., x is odd}
(iv)

(a) {28, 30, 32, 34, 36, 38}
(b) {x : 27 ≤ x.≤  ; x is even}
(v)

(a) number greater than 0 but less than 1
(b) there are infinite numbers between 0 and 1 so the given set can’t be written in Roster form.
(vi) Since there are dark dots at x = – 3, 1, 5 on real line  In roster form, solution set will be {1, – 3, 5}
(vii) {x : x = – 3n and n ∈ N}
(viii) Since there is only one dark dot at x = 2 on real line.
Solution in set builder form is { x | x = 2}
(ix) {x : x = 8 + 2}
(x) {x : x < 5 or 5 < x ∈  10}
(xi) Given y = 5x – 2, x ∈  N
When x = 1 Þ y = 3 ∈  N
When x = 2 Þ y = 10 – 2 = 8 ∈  N
When x = 3 Þ y = 15 – 2 = 13 ∈ N
and so on Thus in roster form, given set = {3, 8, 13, ….}

(xii)

………………………

Question-6

. State whether each 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 number which are multiple of 5.
(iv) The set of animals living on earth.
(v) The set of circles through the origin (0, 0)
(vi) The set of whole numbers greater than 5.
(vii) The set of natural numbers less than one billion.
(viii) The set of integers between – 4 and 4.
(ix) The set of rational numbers between 0 and 1.

Answer-6

(i) there are infinite number of lines which are parallel to x-axis and this counting never comes to end, hence given set is an infinite set.
(ii)  there are 26 English alphabets so the given set is a finite set.
(iii) {5, 10, 15, ……}  multiples of 5 cannot comes to end so given set represents an infinite set.
(iv)  counting of animals living on earth comes to an end so given set is a finite set.
(v) Since there are infinite no. of circles that pass through the origin (0, 0). Hence given set represents an infinite set.
(vi) {6, 7, 8, 9, 10, ……}  whole numbers which are greater than 5 never comes to end.So given set represents an infinite set.

(vii)  natural numbers which are less than one billion are countable so given set represents a finite set.
(viii) {– 3, – 2, – 1, 0, 1, 2, 3}there are 7 integers between – 4 and 4. Thus given set represents a finite set.
(ix)  there may be infinite  rational numbers between 0 and 1.So given set represents an infinite set

Question-7

Which of the following sets are empty sets ?
(i) A = {x : x is a human being living on Mars}
(ii) B = {x : x is an odd number divisible by 2}
(iii) C = {x : x is a point common to any two parallel lines}
(iv) D = {0}
(v) E = {x : x is a natural number, x < 5 and simultaneously x > 7}

Answer-7

(i)  there is no human being living on Mars so set A contains no element. hence given set A be an empty set.
(ii) Since there is no odd number which is divisible by 2 therefore set has no element.  set B be an empty set.
(iii) if two lines are parallel then they can’t be intersect each other so there is no common point between the parallel lines and hence given set C has no element and hence set C be an empty set.
(iv) Given D = {0}. Thus set D contains one element namely 0 set D contains one element and so given set be a non-empty set.
(v) Since there is no natural number which is less than 5 but greater than 7.set contains no element and hence set E be an empty set.

Question-8

Are the following sets equal ? Give reasons.

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

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

Answer-8

(i)

Given A = {2, 3} , set A contains two elements 2 and 3
Now x2 + 5x + 6 = 0
(x + 2) (x + 3) = 0
x = – 2, – 3
B = {– 2, – 3} set B contains two elements – 2 and – 3
because every element of A is not an element of B. Thus A and B are not equal sets.
(ii)

 A = {F, O, L, W}
B = {W, O, L, F}
there A and B have same elements therefore both sets are equal sets.

Question-9

Which of the following are singleton sets ?
(i) A = {x : | x | = 5, x ∈ N}
(ii) A = {Planets of our solar system}
(iii) C = {x : x³ = – 125, x ∈  Z}

Answer-9

(i) Given A = {x : | x | = 5, x ∈ N}
since | x | = 5  x = ± 5, x ∈ N   x = 5
Thus, set A contains only one element ‘5’ set A is a singleton set.
(ii) Given B = {x : x2 – 11x + 24 = 0 ; x ∈ N}
Since x2 – 11x + 24 = 0
x2 – 3x – 8x + 24 = 0
x (x – 3) – 8 (x – 3) = 0
(x – 3) (x – 8) = 0
x = 3, 8 ∈ N
In roster form, B = {3, 8}
So set B contains two elements and hence set B is not a singleton set.
(iii) C = {x : x3 = – 125 ; x ∈ Z} since x³ = – 125 = (– 5)³

 x = – 5 ∈ Z
so set C contains only one element and therefore set C is a singleton set.

Question-10

State the value of n (A) for each of the
following sets.
(i) A = {Months of the year}
(ii) A = {Planets of our solar system}
(iii) A = {x : x is an integer and – 8 ≤ x ≤ 3}
(iv) A = {x : x is an even number}

Answer-10

(i)  there are 12 months in a year  n (A) = no. of elements in set A = 12
(ii) Since there are 8 planets in our solar system.  n (A) = 8
(iii) A = {– 8, – 7, – 6, – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3} . so  n (A) = 12
(iv) A = {……. – 4, – 2, 0, 2, 4, –}  counting of even numbers is never comes to an end and hence A be an infinite set.

TYPE III. (Questions related to interval notation)

Use interval notation to represent each set of numbers

Question-11

. (i) – 17 < x < 0
(ii) 6 ≤ x ≤ 12
(iii) – 1 < x ≤ 4
(iv) – 4 ≤ x < 7
(v) x ≤ 3 or 5 < x ≤ 9
(vi) {x | x ≥ 99}
(vii) {x | x ≠1}
(viii) {1, 3, 5, 7, …}
(ix) x ≠ 3

Answer-11

(i) – 17 < x < 0 = (– 17, 0)
since – 17 and 0 both are not included.
(ii) 6≤  x ≤  12 = [6, 12], since both numbers
6 and 12 are included.
(iii) – 1 < x ≤  4 = (– 1, 4] since 4 is included
and – 1 is not included.
(iv) – 4 ≤  x < 7 = [– 4, 7) Here – 4 is included
but 7 is excluded.
(v) x ≤  3 or 5 < x ≤  9
x ∈ (–  infinite  ,3] ∪ (5, 9]

(vi) {x | x ≤ 99}  x ∈ [99, infinite)
(vii) {x | x 1} = {x | x < 1 or x > 1}
= (– infinite, 1) È (1, infinite)
(viii) {1, 3, 5, 7, …} it can’t be expressible in
interval notation.
(ix) {x | x 3} = {x | x < 3 or x > 3} = (– infinite, 3) ∪ (3, infinite)

Question-12

………..for Question / figure refer textbook………………..

Answer-12

(i) since there is a dark line start at x = – 1 and continuously goes on left side of – 1 therefore Solution set = (– infinity, – 1]

(ii) since dark line starts from 2 and continuously going on right side of x = 2 but there is a hollow dot at x = 2 so 2 is not included. therefore given representation = (2, infinity)

(iii) There is a line between x = 1 and x = 4 and hollow dots at x = 1 and x = 4 shows that 2 and 4 are not included.therefore  given representation = (1, 4)
(iv) There is a dark line from x = – 1 to x = 3 and dark dots at x = – 1 and x = 3 shows that – 1, 3 are included therefore given representation = [– 1, 3]
(v) There is a dark line from x = – 4 to x = – 1 and dark dots at x = – 4 and – 1 shows that – 4 and – 1 included. therefore given representation = [– 4, – 1]


 Exe-1 (b),

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1.

Page-Number 1-10

Question-1

1. Find the subsets of
(i) (a)

(ii) {Reena, Sonu}

(iii) Φ
(iv) {5, {7}}

Answer-1

(i) Required subsets are ; Φ, {a}
(ii) Required subsets are ; Φ, {Sonu}, {Reena}, {Sonu, Reena}
(iii) Required subsets are ; Φ
(iv) Required subsets are ; Φ, {5}, {{7}}, {5, {7}}

Question-2

Let A = {p, q, r}
(i) List all the subsets of A.
(ii) List all the proper subsets of A.

Answer-2

A = {p, q, r}
(i) Φ {p}, {q}, {r}, {p, q}, {q, r}, {p, r}, {p, q, r}
(ii) Φ, {p}, {q}, {r}, {p, q}, {q, r}, {p, r}

Question-3

. Let P = {whole numbers less than 30}
(i) List the subsets Q {even numbers}
(ii) List the subset R {odd numbers}
(iii) List the subset S {prime numbers}
(iv) List the subset T {square numbers}

(v) List the subset U {triangle numbers}
Answer-3

P = {0, 1, 2, 3, ….., 29}
(i) Q = {0, 2, 4, 6, ….., 28}
(ii) R = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
(iii) S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
(iv) T = {0, 1, 4, 9, 16, 25}
(v) U = {0, 1, 3, 6, 10, 15, 21, 28}

Tn= n (n+1) /2

Question-4

4. Tell in each of the following, whether first set is a subset of the second set or not.
(i) A = Set of letters in the word ‘LATE’ B = Set of letters in the word ‘PLATE’

(ii) P = Set of even prime numbers. Q = {x | x = 2p, p Î N and 1 £ p £ 3}
(iii) L = Set of digits in the number 1590 M = Set of digits in the number
178902
(iv) E = Set of all triangles having 4 sides. F = Set of digits in the number ‘100’.

Answer-4

(i) A = {L, A, T, E} and B = {P, L, A, T, E} so every element of set A is a member of set B. therefore A be a subset of B.
(ii)  P = {2} because 2 be the only even prime number and Q = {x | x = 2p, p ∈ N and 1 ≤ p ≤ 3} Since p ∈ N and 1£ p ≤ 3

p={1,2, 3}
When p = 1 then x = 2 × 1 = 2
When p = 2 then x = 2 × 2 = 4
When p = 3 then x = 2 × 3 = 6
hence Q = {2, 4, 6}
every member of set A is a member of set B. hence A is a subset of B.
(iii) L = {0, 1, 5, 9} and M = {0, 1, 2, 7, 8, 9}
since 5 ∈ L but 5 ∉ M. therefore L is not a subset of M.

(iv) E = set of all triangles having 4 sides
since their is no triangle having four sides
E = Φ and F = {0, 1} Clearly E is a subset of F as empty set is a subset of every set.

Question-5

Write the proper subsets of the following sets :
(i) {7} (ii) {1, 3} (iii) {c, a, b} (iv) Φ

Answer-5

(i) Φ
(ii) Φ {1}, {3}
(iii) Φ {c}, {a}, {b}, {c, a}, {a, b}, {c, b}
(iv) no proper subset

Question-6

How many subsets do the following sets have ?
(i) A set having 5 elements.
(ii) The set of letters of the word ‘CENTENARY’

Answer-6

(i) We know that a set having n elements has 2n subsets.
given set having 5 elements 25 of subsets of given set be -. 32.
(ii) Given set = {C, E, N, T, A, R, Y}
Clearly no. of elements in given set be 7.
…2n..=..27….= 128…………
Question-7

How many proper subsets do the following sets have ?
(i) The set of factors of 12.
(ii) The set A {x | x is a prime numbers, x < 20}

Answer-7

We know that if a set A has n elements. Then number of proper subsets of A = ..2n.- 1………….
(i) A = set of all factors of 12 = {1, 2, 3, 4,6, 12}
there 6 element so 2n.- 1..= 26.- 1..=64-1 = 63
(ii) A = {2, 3, 5, 7, 11, 13, 17, 19} there n = 8
there 8 element so 2n.- 1..= 28.- 1..=256-1 = 255

Question-8

Answer true or false
(i) ……………..
(iii) …………………
(v) ………………………
(vii) For any two sets A and B either ……………….
(viii) Every set has a proper subset.
(ix) Every subset of a finite set is finite.
(x) Every subset of an infinite set is infinite

Answer-8

(i) 3 ∈ {3, 0} but 3 is not a subset of {3, 0}
false.
(ii) since {3} is a subset of {3, 0}
true.
(iii) since f is not a member of {3, 0} Ø ∉ {3, 0}
false.
(iv) Since 0 be a member of {3, 0}
0 ∈ {3, 0}
true.

(v) Since f i.e. empty set is proper subset of {3, 0}.
true.
(vi) empty set is a subset of every set.
true.
(vii)  A = {1, 2} and B = {4, 5}
False,
(viii) Since empty set has no proper subset
false.
(ix) True statement
(x) since empty set is a subset of
every set and f is a finite set
False,

Question-9

Find the power set of each of the following
sets :
(i) A = {digits in the number 98}

(ii) B = {letters in the word ‘KID’}
(iii) S = {2, 3}
(iv) T = {4, 7, 9}

Answer-9

power set is the set of all subsets of given set.
(i) Given A = {8, 9}
P (A) = {Φ, {8}, {9}, {8, 9}}
(ii) B = {K, I, D}
P (A) = {Φ, {K}, {I}, {D}, {K, I}, {K, D}, {I, D}, {K, I, D}}
(iii) S = {2, 3}
P (S) = {Φ, {2}, {3}, {2, 3}}
(iv) T = {4, 7, 9}
P (T) = {Φ, {4}, {7}, {9}, {4, 7}, {4, 9}, {7, 9}, {4, 7, 9}}


 Exe-1 (c),

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1.

Page-Number 1-17


 Exe-1 (d),

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1.

Page-Number 1-23


Chapter Test

Sets OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-1.

Page-Number 1-29

-: End of Sets OP Malhotra S. Chand ISC Class-11 Maths Chapter-1 Solution :-

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