Class 11 Maths - Sets Chapter
CBSE Class 11 Maths Sets, Sets Introduction, Sets Representations, Empty Set, Finite Sets, Infinite Sets, Equal Sets, Subsets, Universal Set, Venn Diagrams, Operations on Sets, Complement of a Set, CBSE Maths Solutions, CBSE Sets Exercise Solutions
Exercise 1.1 - Sets
(i) The collection of all the months of a year beginning with the letter J.
Yes, because it is a well-defined collection of objects.
(ii) The collection of ten most talented writers of India.
No, because it is subjective and not well-defined.
(iii) A team of eleven best-cricket batsmen of the world.
No, because it is subjective and not well-defined.
(iv) The collection of all boys in your class.
Yes, because it is a well-defined collection of objects.
(v) The collection of all natural numbers less than 100.
Yes, because it is a well-defined collection of objects.
(vi) A collection of novels written by the writer Munshi Prem Chand.
Yes, because it is a well-defined collection of objects.
(vii) The collection of all even integers.
Yes, because it is a well-defined collection of objects.
(viii) The collection of questions in this Chapter.
Yes, because it is a well-defined collection of objects.
(ix) A collection of most dangerous animals of the world.
No, because it is subjective and not well-defined.
(i) 5 __ A
∈
(ii) 8 __ A
∉
(iii) 0 __ A
∉
(iv) 4 __ A
∈
(v) 2 __ A
∈
(vi) 10 __ A
∉
(i) A = {x : x is an integer and –3 ≤ x < 7}
A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
B = {1, 2, 3, 4, 5}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
C = {17, 26, 35, 44, 53, 62, 71, 80, 89, 98}
(iv) D = {x : x is a prime number which is divisor of 60}
D = {2, 3, 5}
(v) E = The set of all letters in the word TRIGONOMETRY
E = {T, R, I, G, O, N, M, E, Y}
(vi) F = The set of all letters in the word BETTER
F = {B, E, T, R}
(i) {3, 6, 9, 12}
A = {x : x = 3n, n ∈ N, 1 ≤ n ≤ 4}
(ii) {2, 4, 8, 16, 32}
B = {x : x = 2^n, n ∈ N, 1 ≤ n ≤ 5}
(iii) {5, 25, 125, 625}
C = {x : x = 5^n, n ∈ N, 1 ≤ n ≤ 4}
(iv) {2, 4, 6, . . .}
D = {x : x = 2n, n ∈ N}
(v) {1, 4, 9, . . ., 100}
E = {x : x = n^2, n ∈ N, 1 ≤ n ≤ 10}
(i) A = {x : x is an odd natural number}
A = {1, 3, 5, 7, 9, 11, 13, 15, ...}
(ii) B = {x : x is an integer, 1/2 < x < 9/2}
B = {1, 2, 3, 4}
(iii) C = {x : x is an integer, x^2 ≤ 4}
C = {-2, -1, 0, 1, 2}
(iv) D = {x : x is a letter in the word “LOYAL”}
D = {L, O, Y, A}
(v) E = {x : x is a month of a year not having 31 days}
E = {February, April, June, September, November}
(vi) F = {x : x is a consonant in the English alphabet which precedes k}
F = {b, c, d, f, g, h, j}
(i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6}
c
(ii) {2, 3} (b) {x : x is an odd natural number less than 10}
a
(iii) {M, A, T, H, E, I, C, S} (c) {x : x is natural number and divisor of 6}
d
(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word “MATHEMATICS”}
b
Exercise 1.2 - Sets
1. Which of the following are examples of the null set:
(i) Set of odd natural numbers divisible by 2
(i), (iii), (iv)
(ii) Set of even prime numbers
Not a null set (contains the number 2)
(iii) { x : x is a natural number, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}
2. Which of the following sets are finite or infinite:
(i) The set of months of a year
Finite
(ii) {1, 2, 3, . . .}
Infinite
(iii) {1, 2, 3, . . .99, 100}
Finite
(iv) The set of positive integers greater than 100
Infinite
(v) The set of prime numbers less than 99
Finite
3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
Infinite
(ii) The set of letters in the English alphabet
Finite
(iii) The set of numbers which are multiple of 5
Infinite
(iv) The set of animals living on the earth
Finite
(v) The set of circles passing through the origin (0,0)
Infinite
4. In the following, state whether A = B or not:
(i) A = { a, b, c, d } B = { d, c, b, a }
A = B
(ii) A = { 4, 8, 12, 16 } B = { 8, 4, 16, 18}
A ≠ B
(iii) A = {2, 4, 6, 8, 10} B = { x : x is positive even integer and x ≤ 10}
A = B
(iv) A = { x : x is a multiple of 10}, B = { 10, 15, 20, 25, 30, . . . }
A ≠ B
5. Are the following pair of sets equal? Give reasons:
(i) A = {2, 3}, B = {x : x is solution of x2 + 5x + 6 = 0}
A = B (Solutions are 2 and 3)
(ii) A = { x : x is a letter in the word FOLLOW} B = { y : y is a letter in the word WOLF}
A = B (Both sets contain the same letters: F, O, L, W)
6. From the sets given below, select equal sets:
A = { 2, 4, 8, 12}, B = { 1, 2, 3, 4}, C = { 4, 8, 12, 14}, D = { 3, 1, 4, 2}
E = {–1, 1}, F = { 0, a}, G = {1, –1}, H = { 0, 1}
B = D, E = G
Exercise 1.3
1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:
(i) { 2, 3, 4 } ⊂ { 1, 2, 3, 4,5 }
⊂
(ii) { a, b, c } ⊄ { b, c, d }
⊄
(iii) {x : x is a student of Class XI of your school} ⊂ {x : x student of your school}
⊂
(iv) {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit}
⊄
(v) {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}
⊄
(vi) {x : x is an equilateral triangle in a plane} ⊂ {x : x is a triangle in the same plane}
⊂
(vii) {x : x is an even natural number} ⊂ {x : x is an integer}
⊂
2. Examine whether the following statements are true or false:
(i) { a, b } ⊄ { b, c, a }
False (a set is a subset of itself)
(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}
True
(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }
False
(iv) { a } ⊂ { a, b, c }
True
(v) { a } ∈ { a, b, c }
False
(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36}
True
3. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
Incorrect (3, 4 is an element of A, not a subset)
(ii) {3, 4} ∈ A
Correct
(iii) {{3, 4}} ⊂ A
Correct
(iv) 1 ∈ A
Correct
(v) 1 ⊂ A
Incorrect (1 is an element of A, not a subset)
(vi) {1, 2, 5} ⊂ A
Correct
(vii) {1, 2, 5} ∈ A
Incorrect (it is a subset, not an element)
(viii) {1, 2, 3} ⊂ A
Incorrect
(ix) φ ∈ A
Incorrect (empty set is a subset, not an element)
(x) φ ⊂ A
Correct
(xi) {φ} ⊂ A
Correct
4. Write down all the subsets of the following sets:
(i) {a}
φ, {a}
(ii) {a, b}
φ, {a}, {b}, {a, b}
(iii) {1, 2, 3}
φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}
(iv) φ
φ
5. Write the following as intervals:
(i) {x : x ∈ R, – 4 < x ≤ 6}
(–4, 6]
(ii) {x : x ∈ R, – 12 < x < –10}
(–12, –10)
(iii) {x : x ∈ R, 0 ≤ x < 7}
[0, 7)
(iv) {x : x ∈ R, 3 ≤ x ≤ 4}
[3, 4]
6. Write the following intervals in set-builder form:
(i) (– 3, 0)
{x : x ∈ R, –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}
7. What universal set(s) would you propose for each of the following:
(i) The set of right triangles
The set of all triangles
(ii) The set of isosceles triangles
The set of all triangles
8. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set(s) for all the three sets A, B and C:
(i) {0, 1, 2, 3, 4, 5, 6}
Yes
(ii) φ
No
(iii) {0,1,2,3,4,5,6,7,8,9,10}
Yes
(iv) {1,2,3,4,5,6,7,8}
No
Exercise 1.4
1. Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5}, Y = {1, 2, 3}
X ∪ Y = {1, 2, 3, 5}
(ii) A = {a, e, i, o, u}, B = {a, b, c}
A ∪ B = {a, e, i, o, u, b, c}
(iii) A = {x : x is a natural number and multiple of 3}, B = {x : x is a natural number less than 6}
A ∪ B = {1, 2, 3, 6, 9}
(iv) A = {x : x is a natural number and 1 < x ≤ 6}, B = {x : x is a natural number and 6 < x < 10}
A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
(v) A = {1, 2, 3}, B = φ
A ∪ B = {1, 2, 3}
2. Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
A ⊂ B (Yes, A is a subset of B)
A ∪ B = {a, b, c}
3. If A and B are two sets such that A ⊂ B, then what is A ∪ B?
A ∪ B = B (Union of A and B is B)
4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, and D = {7, 8, 9, 10}; find:
(i) A ∪ B = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
5. Find the intersection of each pair of sets of question 1 above.
(i) X ∩ Y = {1, 3}
(ii) A ∩ B = {a}
(iii) A ∩ B = φ
(iv) A ∩ B = {6}
(v) A ∩ B = φ
6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, and D = {15, 17}; find:
(i) A ∩ B = {7, 9, 11}
(ii) B ∩ C = {7, 9, 11, 13}
(iii) A ∩ C ∩ D = φ
(iv) A ∩ C = {11}
(v) B ∩ D = φ
(vi) A ∩ (B ∪ C) = {3, 5, 7, 9, 11, 13}
(vii) A ∩ D = φ
(viii) A ∩ (B ∪ D) = {3, 5, 7, 9, 11, 13}
(ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11}
(x) (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 13}
7. If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number}, and D = {x : x is a prime number}, find:
(i) A ∩ B = {2, 4, 6, 8, ...}
(ii) A ∩ C = {1, 3, 5, 7, ...}
(iii) A ∩ D = {2, 3, 5, 7, 11, ...}
(iv) B ∩ C = φ
(v) B ∩ D = {2}
(vi) C ∩ D = {3, 5, 7, 11, ...}
8. Which of the following pairs of sets are disjoint?
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}
Disjoint (No common elements)
(ii) {a, e, i, o, u} and {c, d, e, f}
Not disjoint (Common element: e)
(iii) {x : x is an even integer} and {x : x is an odd integer}
Disjoint (No common elements)
9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find:
(i) A − B = {3, 6, 9, 15, 18, 21}
(ii) A − C = {3, 9, 15, 18, 21}
(iii) A − D = {3, 6, 9, 12, 18, 21}
(iv) B − A = {4, 8, 16, 20}
(v) C − A = {2, 4, 6, 8, 10, 14, 16}
(vi) D − A = {5, 10, 20}
(vii) B − C = {16, 20}
(viii) B − D = {4, 8}
(ix) C − B = {2, 6, 10, 14}
(x) D − B = {5, 10, 15}
(xi) C − D = {2, 6, 8, 10, 14}
(xii) D − C = {5}
10. If X = {a, b, c, d} and Y = {f, b, d, g}, find:
(i) X − Y = {a, c}
(ii) Y − X = {f, g}
(iii) X ∩ Y = {b, d}
11. If R is the set of real numbers and Q is the set of rational numbers, then what is R − Q?
R − Q = Set of irrational numbers
12. 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.
False (Common element: 3)
(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
False (Common element: a)
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
True (No common elements)
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
True (No common elements)
Exercise 1.5
1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, and C = {3, 4, 5, 6}. Find:
(i) A′ = {5, 6, 7, 8, 9}
(ii) B′ = {1, 3, 5, 7, 9}
(iii) (A ∪ C)′ = {5, 7, 8, 9}
(iv) (A ∪ B)′ = {5, 7, 9}
(v) (A′)′ = {1, 2, 3, 4}
(vi) (B − C)′ = {1, 3, 5, 7, 9}
2. If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
(i) A = {a, b, c}
A′ = {d, e, f, g, h}
(ii) B = {d, e, f, g}
B′ = {a, b, c, h}
(iii) C = {a, c, e, g}
C′ = {b, d, f, h}
(iv) D = {f, g, h, a}
D′ = {b, c, d, e}
3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x : x is an even natural number}
{x : x is an odd natural number}
(ii) {x : x is an odd natural number}
{x : x is an even natural number}
(iii) {x : x is a positive multiple of 3}
{x : x is not a positive multiple of 3}
(iv) {x : x is a prime number}
{x : x is a composite number}
(v) {x : x is a natural number divisible by 3 and 5}
{x : x is not a natural number divisible by 3 and 5}
(vi) {x : x is a perfect square}
{x : x is not a perfect square}
(vii) {x : x is a perfect cube}
{x : x is not a perfect cube}
(viii) {x : x + 5 = 8}
{x : x + 5 ≠ 8}
(ix) {x : 2x + 5 = 9}
{x : 2x + 5 ≠ 9}
(x) {x : x ≥ 7}
{x : x < 7}
(xi) {x : x ∈ N and 2x + 1 > 10}
{x : x ∈ N and 2x + 1 ≤ 10}
4. If 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 ∪ B)′ = A′ ∩ B′
Left side: {1, 9}
Right side: {1, 9}
Both sides are equal.
(ii) (A ∩ B)′ = A′ ∪ B′
Left side: {1, 3, 5, 7, 9}
Right side: {1, 3, 5, 7, 9}
Both sides are equal.
5. Draw appropriate Venn diagrams for each of the following:
(i) (A ∪ B)′
(ii) A′ ∩ B′
(iii) (A ∩ B)′
(iv) A′ ∪ B′
(i) (A ∪ B)′
_______________________
/ \
| A ∪ B |
\_______________________/
/ Universal set \
| (A ∪ B)′ = Elements outside A ∪ B |
\__________________________________/
(ii) A′ ∩ B′
_______________________
/ \
| A′ |
| _________ ________|
\ / \ / /
| \/ /
| A′ ∩ B′ | /
| /\________/
| / \
\_________/ \_______
/ \
| Universal set |
\_______________________/
(iii) (A ∩ B)′
_______________________
/ \
| A ∩ B |
\_______________________/
/ Universal set \
| (A ∩ B)′ = Elements outside A ∩ B |
\__________________________________/
(iv) A′ ∪ B′
_______________________
/ \
| A′ |
| _________ ________|
\ / \ / /
| \/ /
| A′ ∪ B′ | /
| /\________/
| / \
\_________/ \_______
/ \
| Universal set |
\_______________________/
6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
A′ = Set of all triangles with all angles 60°
7. Fill in the blanks to make each of the following a true statement:
(i) A ∪ A′ = _______
Universal set (U)
(ii) A ∩ A′ = _______
Empty set (φ)
(iii) (A ∪ B) ∩ A′ = _______
B ∩ A′
(iv) (A ∩ B) ∪ A′ = _______
A′ ∪ B
Miscellaneous Exercise on Chapter 1
1. Decide, among the following sets, which sets are subsets of one another:
- A = { x : x ∈ ℝ and x satisfy x^2 – 8x + 12 = 0 }
- B = { 2, 4, 6 }
- C = { 2, 4, 6, 8, ... }
- D = { 6 }
2. Determine whether the following statements are true or false:
- (i) If x ∈ A and A ⊆ B, then x ∈ B.
- (ii) If A ⊂ B and B ∈ C, then A ∈ C.
- (iii) If A ⊂ B and B ⊂ C, then A ⊂ C.
- (iv) If A ⊄ B and B ⊄ C, then A ⊄ C.
- (v) If x ∈ A and A ⊄ B, then x ∈ B.
- (vi) If A ⊂ B and x ∉ B, then x ∉ A.
3. Let A, B, and C be sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.
- Since A ∪ B = A ∪ C, and A ∩ B = A ∩ C,
- Therefore, B = C by the properties of unions and intersections.
4. Show that the following four conditions are equivalent:
- (i) A ⊂ B
- (ii) A − B = φ
- (iii) A ∪ B = B
- (iv) A ∩ B = A
5. Show that if A ⊂ B, then C − B ⊂ C − A.
- Since A ⊂ B, any element in B that is not in A is also not in C − A.
- Therefore, C − B ⊂ C − A.
6. Show that for any sets A and B,
- A = (A ∩ B) ∪ (A − B)
- A ∪ (B − A) = (A ∪ B)
7. Using properties of sets, show that
- (i) A ∪ (A ∩ B) = A
- (ii) A ∩ (A ∪ B) = A
8. Show that A ∩ B = A ∩ C need not imply B = C.
- Let A = {1}, B = {1, 2}, C = {1, 3}.
- Then A ∩ B = A ∩ C = {1}, but B ≠ C.
9. Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.
- Since A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X,
- Therefore, A = A ∩ (A ∪ X) = B = B ∩ (B ∪ X) = B.
10. Find sets A, B, and C such that A ∩ B, B ∩ C, and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.
- Let A = {1, 2}, B = {2, 3}, C = {3, 4}.
- Then A ∩ B = {2}, B ∩ C = {3}, A ∩ C = ∅, and A ∩ B ∩ C = ∅.
