MATHEMATICS CLASS 11
Class 11 - Chapter 1: Sets
NCERT Solutions for Class 11 Mathematics Chapter 1 Sets
Exercise 1.4
Question 1
Find the union of each of the following pairs of sets.
(i)
X = {1, 3, 5}
Y = {1, 2, 3}
Solution:
X ∪ Y is the set of all elements belonging to X or Y.
X ∪ Y = {1, 2, 3, 5}
(ii)
A = {a, e, i, o, u}
B = {a, b, c}
A ∪ B = {a, b, c, e, i, o, u}
(iii)
A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
A = {3, 6, 9, 12, ...}
B = {1, 2, 3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5, 6, 9, 12, ...}
(iv)
A = {x : x is a natural number and 1 < x ≤ 6}
B = {x : x is a natural number and 6 < x < 10}
A = {2, 3, 4, 5, 6}
B = {7, 8, 9}
A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
(v)
A = {1, 2, 3}
B = φ
A ∪ B = A
A ∪ B = {1, 2, 3}
Question 2
Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
Solution:
Every element of A belongs to B.
Therefore,
A ⊂ B
A ∪ B = {a, b, c}
Question 3
If A and B are two sets such that A ⊂ B, then what is A ∪ B?
Solution:
Since every element of A already belongs to B,
A ∪ B = B
Question 4
If
A = {1, 2, 3, 4}
B = {3, 4, 5, 6}
C = {5, 6, 7, 8}
D = {7, 8, 9, 10}
Find the following:
| Part | Answer |
|---|---|
| (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} |
Question 5
Find the intersection of each pair of sets of Question 1.
| Part | Answer |
|---|---|
| (i) X ∩ Y | {1, 3} |
| (ii) A ∩ B | {a} |
| (iii) A ∩ B | {3} |
| (iv) A ∩ B | φ |
| (v) A ∩ φ | φ |
Question 6
Let
A = {3, 5, 7, 9, 11}
B = {7, 9, 11, 13}
C = {11, 13, 15}
D = {15, 17}
| Part | Answer |
|---|---|
| (i) A ∩ B | {7, 9, 11} |
| (ii) B ∩ C | {11, 13} |
| (iii) A ∩ C ∩ D | φ |
| (iv) A ∩ C | {11} |
| (v) B ∩ D | φ |
| (vi) A ∩ (B ∪ C) | {7, 9, 11} |
| (vii) A ∩ D | φ |
| (viii) A ∩ (B ∪ D) | {7, 9, 11} |
| (ix) (A ∩ B) ∩ (B ∪ C) | {7, 9, 11} |
| (x) (A ∪ D) ∩ (B ∪ C) | {7, 9, 11, 13, 15} |
Question 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}
D = {x : x is a prime number}
| Part | Answer |
|---|---|
| (i) A ∩ B | B |
| (ii) A ∩ C | C |
| (iii) A ∩ D | D |
| (iv) B ∩ C | φ |
| (v) B ∩ D | {2} |
| (vi) C ∩ D | {3, 5, 7, 11, ...} |
Question 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}
Intersection = {4}
Hence, not disjoint.
(ii) {a, e, i, o, u} and {c, d, e, f}
Intersection = {e}
Hence, not disjoint.
(iii) {x : x is an even integer} and {x : x is an odd integer}
Intersection = φ
Hence, they are disjoint sets.
Question 9
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}
| Part | Answer |
|---|---|
| (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, 8, 10, 14, 16} |
| (vi) D − A | {5, 10, 20} |
| (vii) B − C | {20} |
| (viii) B − D | {4, 8, 12, 16} |
| (ix) C − B | {2, 6, 10, 14} |
| (x) D − B | {5, 10, 15} |
| (xi) C − D | {2, 4, 6, 8, 12, 14, 16} |
| (xii) D − C | {5, 15, 20} |
Question 10
X = {a, b, c, d}
Y = {f, b, d, g}
(i) X − Y = {a, c}
(ii) Y − X = {f, g}
(iii) X ∩ Y = {b, d}
Question 11
If R is the set of real numbers and Q is the set of rational numbers, then
R − Q = Set of all irrational numbers.
Question 12
State whether each statement is true or false.
| Statement | Answer |
|---|---|
| (i) {2,3,4,5} and {3,6} are disjoint sets. | False |
| (ii) {a,e,i,o,u} and {a,b,c,d} are disjoint sets. | False |
| (iii) {2,6,10,14} and {3,7,11,15} are disjoint sets. | True |
| (iv) {2,6,10} and {3,7,11} are disjoint sets. | True |