1. Sets Mathematics Class 11 In English Medium Ncert Book Solutions Exercise 1.5
1. Sets : Exercise 1.5 Mathematics class 11th:English Medium NCERT Book Solutions
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1. Sets
Exercise 1.5
Exercise 1.5
Q1. 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′
(ii) B′
(iii) (A ∪ C)′
(iv) (A ∪ B)′
(v) (A′)′
(vi) (B – C)′
Solution: Given that
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 }.
(i) A' = {5, 6, 7, 8, 9}
(ii) B' = {1, 3, 5, 7, 9}
(iii) A ∪ C = {1, 2, 3, 4, 5, 6}
Therefore, (A ∪ C)′ = {7, 8, 9}
(iv) A ∪ B = {1, 2, 3, 4, 6, 8}
Therefore, (A ∪ B)′ = {5, 7, 9}
(v) A' = {5, 6, 7, 8, 9}
(A')' = A = {1, 2, 3, 4}
(vi) B - C = {2, 8}
(B - C)' = 1, 3, 4, 5, 6, 7, 9}
Q2. If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = { f, g, h, a}
Solution: Given that
U = { a, b, c, d, e, f, g, h}
(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}
Q3. 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}
(ii) { x : x is an odd natural number }
(iii) {x : x is a positive multiple of 3}
(iv) { x : x is a prime number }
(v) {x : x is a natural number divisible by 3 and 5}
(vi) { x : x is a perfect square }
(vii) { x : x is a perfect cube}
(viii) { x : x + 5 = 8 }
(ix) { x : 2x + 5 = 9}
(x) { x : x ≥ 7 }
(xi) { x : x ∈ N and 2x + 1 > 10 }
Solution: Given that U = { 1, 2, 3, 4, 5, 6, 7 ....}
(i) Let A = {x : x is an even natural number}
Or A = {2, 4, 6, 8 .....}
A' = { 1, 3, 5, 7 .....}
= {x : x is an odd natural number}
(ii) Let B = { x : x is an odd natural number }
Or B = { 1, 3, 5, 7 .....}
B' = {2, 4, 6, 8 .....}
= {x : x is an even natural number}
(iii) Let C = {x : x is a positive multiple of 3}
Or C = {3, 6, 9 ....}
C' = {1, 2, 4, 5, 7, 8, 10 .....}
= {x: x N and x is not a multiple of 3}
(iv) Let D = { x : x is a prime number }
Or D = {2, 3, 5, 7, 11 ... }
D' = {1, 4, 6, 8, 9, 10 ...... }
= {x: x is a positive composite number and x = 1}
(v) Let E = {x : x is a natural number divisible by 3 and 5}
Or E = {15, 30, 45 .....}
E' = {x: x is a natural number that is not divisible by 3 or 5}
(vi) Let F = { x : x is a perfect square }
F' = {x: x N and x is not a perfect square}
(vii) Let G = {x: x is a perfect cube}
G' = {x: x N and x is not a perfect cube}
(viii) Let H = {x: x + 5 = 8}
H' = {x: x N and x ≠ 3}
(ix) Let I = {x: 2x + 5 = 9}
I' = {x: x N and x ≠ 2}
(x) Let J = {x: x ≥ 7}
J' = {x: x N and x < 7}
(xi) Let K = {x: x N and 2x + 1 > 10}
K = {x: x N and x ≤ 9/2}
Q4. 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′
(ii) (A ∩ B)′ = A′ ∪ B′
Solution:
(i) U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}.
(A ∪ B)′ = A′ ∩ B′
A ∪ B = {2, 3, 4, 5, 6, 7, 8}
LHS = (A ∪ B)′ = {1, 9} ...(i)
RHS = A′ ∩ B′
= {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
= {1, 9} .... (ii)
LHS = RHS
Hence Verified.
Solution:
(ii) U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}.
(A ∩ B)′ = A′ ∪ B′
A ∩ B = {2}
LHS = (A ∩ B)′ = {1, 3, 4, 5, 6, 7, 8, 9 }
RHS = A′ ∪ B′
= {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9}
= {1, 3, 4, 5, 6, 7, 8, 9 }
LHS = RHS
Hence Verified
Q5. Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)′,
(ii) A′ ∩ B′,
(iii) (A ∩ B)′,
(iv) A′ ∪ B′
Solution:
(i) (A ∪ B)′
Venn diagram of (A ∪ B)′
(ii) A′ ∩ B′,
Venn diagram of A′ ∩ B′
Note: Venn diagram of A′ ∩ B′ will be same as (A ∪ B)′
Because (A ∪ B)′ = A′ ∩ B′
(iii) (A ∩ B)′
Venn diagram of (A ∩ B)′
(iv) A′ ∪ B′
Venn diagram of A′ ∪ B′
Q6. 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′?
Solution:
A = {the set of all triangles with at least one angle different from 60°}
A' = {the set of all equilateral triangles}
Q7. Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = . . .
(ii) φ′ ∩ A = . . .
(iii) A ∩ A′ = . . .
(iv) U′ ∩ A = . . .
Solution:
(i) A ∪ A′ = U
(ii) φ′ = U
Therefore φ′ ∩ A = U ∩ A = A
so, φ′ ∩ A = A
(iii) A ∩ A′ = φ
(iv) U′ ∩ A = φ
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See other sub-topics of this chapter:
1. Exercise 1.1 class 11 Chap-1. Sets
2. Exercise 1.2 class 11 Chap-1. Sets
3. Exercise 1.3 class 11 Chap-1. Sets
4. Exercise 1.4 class 11 Chap-1. Sets
5. Exercise 1.5 class 11 Chap-1. Sets
6. Exercise 1.6 class 11 Chap-1. Sets
7. Miscellaneous class 11 Chap-1. Sets
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Mathematics Chapter List
1. Sets
2. Relations and Functions
3. Trigonometric Functions
4. Principle Of Mathematical Induction
5. Complex Numbers and Quadratic Equations
6. Linear Inequalities
7. Permutations and Combinations
8. Binomial Theorem
9. Sequences and Series
10. Straight Lines
11. Conic Sections
12. Introduction to Three Dimensional Geometry
13. Limits and Derivatives
14. Mathematical Reasoning
15. Statistics
16. Probability