NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.5
If you're in search for Chapter 1 Sets Exercise 1.5 NCERT Solutions for Class 11 Math then you can find them here. Class 11 Maths NCERT Solutions provided by us will help you in solving difficult questions in easy manner and no time. By practicing these questions, you become aware of important formulas and its application that will help you in higher classes and entrance exams.
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', (ii) B' (iii) (A ∪ C)' (iv) (A ∪ B)' (v) (A')' (vi) (B – C)'
Answer
Here, 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' = U – A = {5, 6, 7, 8, 9}
(ii) B' = U – B = {1, 3, 5, 7, 9}
(iii) A∪ C = {1, 2, 3, 4}∪ {3, 4, 5,6}
= {1, 2, 3, 4, 5, 6}
Hence (A ∪ C)' = U – (A ∪ C) = {7, 8, 9}
(iv) A ∪ B = {1, 2, 3, 4}∪ {2, 4, 6, 8}
= {1, 2, 3, 4, 6, 8}
Hence (A ∪ B)' = U – (A ∪ B) = {5, 7, 9}
(v) A' = U – A = {5, 6, 7, 8, 9}
Hence (A')' = U – A'= {1, 2, 3, 4}
(vi) B – C = {2, 4, 6, 8} – {3, 4, 5, 6} = {2, 8}
Hence (B – C)' = U – (B – C) = {1, 3, 4, 5, 6, 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}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}
Answer
Here, U – {a, b, c, d, e, f, g, h}
(i) A' = U – A = {a, b, c, d,e, f, g, h} – {a, b, c}
= {d, e, g, g, h}
(ii) B' = U – B = {a, b, c, d,e, f, g, h} – {d, e, f, g}
= {a, b, c, h}
(iii) C' = U – C = {a, b, c, d,e, f, g, h} – {a, c, e, g}
= {b, d, f, h}
(iv) D' = U – D = {a, b, c, d,e, f, g, h} – {f, g, h,a}
= {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}
(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 ε N and 2x + 1 > 10}
Answer
(i) {x: x is an odd natural number}
(ii) {x: x is an even natural number}
(iii) {x: x ε N and x is not a multiple of 3}
(iv) {x: x is a positive composite number and x = 1}
(v) {x: x ε n N and x is neither a multiples of 3 nor of 5}.
(vi) {x: x ε N and x is not a perfect square}
(vii) {x: x ε N and x is not a perfect cube}
(viii) {x: x ε N and x ≠ 3}
[x + 5 = 8 ⇒ x = 8 – 5 = 3]
(ix) {x: x ε N and x ≠ 2}
[2x + 5 = 9 ⇒ 2x = 9 – 5 ⇒ 2x = 4 ⇒ x = 2]
(x) {x: x ε N and x < 7} = {1, 2, 3, 4, 5, 6},
(xi) { x : x ε N and x ≤ 9/2}
[2x + 1 > 10 ⇒ 2x > 10 – 1 ⇒ 2x = 9
⇒ x > 9/2]
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'
(ii) (A∩B)' = A' ∪ B'
Answer
(i) A ∪ B = {2, 4, 6, 8} ∪{2, 3, 5, 7}
= {2, 3, 4, 5, 6, 7, 8}
L.H.S. = (A ∪ B)¢ = U – (A ∪ B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 4, 5, 6, 7, 8}
= {1, 9}
Now, A' = U – A
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}
and B' = U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7}
= {1, 4, 6, 8, 9}
R.H.S. = A'∩ B'
= {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1,9}
L.H.S. = R.H.S.
Hence (A∪B)' = A'∩ B' is verified.
(ii) A ∩B = {2, 4, 6, 8} ∩ {2, 3, 5, 7} = {2}
L.H.S. = (A∩ B)' = U – (A∩ B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2}
= {1, 3, 4, 5, 6, 7, 8, 9}
R.H.S. = A' ∪ B'
= {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9}
= {1, 3, 4, 5, 6, 7, 8, 9}
L.H.S = R.H.S.
Hence, (A ∩ B) = A' ∪ B' is verified.
5. Draw appropriate Venn diagram for each of the following:
(i) (A ∪ B)'
(ii) A' ∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'.
Answer
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', (ii) B' (iii) (A ∪ C)' (iv) (A ∪ B)' (v) (A')' (vi) (B – C)'
Answer
Here, 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' = U – A = {5, 6, 7, 8, 9}
(ii) B' = U – B = {1, 3, 5, 7, 9}
(iii) A∪ C = {1, 2, 3, 4}∪ {3, 4, 5,6}
= {1, 2, 3, 4, 5, 6}
Hence (A ∪ C)' = U – (A ∪ C) = {7, 8, 9}
(iv) A ∪ B = {1, 2, 3, 4}∪ {2, 4, 6, 8}
= {1, 2, 3, 4, 6, 8}
Hence (A ∪ B)' = U – (A ∪ B) = {5, 7, 9}
(v) A' = U – A = {5, 6, 7, 8, 9}
Hence (A')' = U – A'= {1, 2, 3, 4}
(vi) B – C = {2, 4, 6, 8} – {3, 4, 5, 6} = {2, 8}
Hence (B – C)' = U – (B – C) = {1, 3, 4, 5, 6, 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}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}
Answer
Here, U – {a, b, c, d, e, f, g, h}
(i) A' = U – A = {a, b, c, d,e, f, g, h} – {a, b, c}
= {d, e, g, g, h}
(ii) B' = U – B = {a, b, c, d,e, f, g, h} – {d, e, f, g}
= {a, b, c, h}
(iii) C' = U – C = {a, b, c, d,e, f, g, h} – {a, c, e, g}
= {b, d, f, h}
(iv) D' = U – D = {a, b, c, d,e, f, g, h} – {f, g, h,a}
= {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}
(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 ε N and 2x + 1 > 10}
Answer
(i) {x: x is an odd natural number}
(ii) {x: x is an even natural number}
(iii) {x: x ε N and x is not a multiple of 3}
(iv) {x: x is a positive composite number and x = 1}
(v) {x: x ε n N and x is neither a multiples of 3 nor of 5}.
(vi) {x: x ε N and x is not a perfect square}
(vii) {x: x ε N and x is not a perfect cube}
(viii) {x: x ε N and x ≠ 3}
[x + 5 = 8 ⇒ x = 8 – 5 = 3]
(ix) {x: x ε N and x ≠ 2}
[2x + 5 = 9 ⇒ 2x = 9 – 5 ⇒ 2x = 4 ⇒ x = 2]
(x) {x: x ε N and x < 7} = {1, 2, 3, 4, 5, 6},
(xi) { x : x ε N and x ≤ 9/2}
[2x + 1 > 10 ⇒ 2x > 10 – 1 ⇒ 2x = 9
⇒ x > 9/2]
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'
(ii) (A∩B)' = A' ∪ B'
Answer
(i) A ∪ B = {2, 4, 6, 8} ∪{2, 3, 5, 7}
= {2, 3, 4, 5, 6, 7, 8}
L.H.S. = (A ∪ B)¢ = U – (A ∪ B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 4, 5, 6, 7, 8}
= {1, 9}
Now, A' = U – A
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}
and B' = U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7}
= {1, 4, 6, 8, 9}
R.H.S. = A'∩ B'
= {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1,9}
L.H.S. = R.H.S.
Hence (A∪B)' = A'∩ B' is verified.
(ii) A ∩B = {2, 4, 6, 8} ∩ {2, 3, 5, 7} = {2}
L.H.S. = (A∩ B)' = U – (A∩ B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2}
= {1, 3, 4, 5, 6, 7, 8, 9}
R.H.S. = A' ∪ B'
= {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9}
= {1, 3, 4, 5, 6, 7, 8, 9}
L.H.S = R.H.S.
Hence, (A ∩ B) = A' ∪ B' is verified.
5. Draw appropriate Venn diagram for each of the following:
(i) (A ∪ B)'
(ii) A' ∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'.
Answer
(i) (A ∪ B)'
The shaded region indicates (A ∪ B)'
(ii) A' ∩ B'
The shaded region indicates A' ∩ B'
(iii) (A ∩ B)'
The shaded region indicates (A ∩ B)'
(iv) A' ∪ B'
The shaded region indicates A' ∪ B'
All shaded region formed by all horizontal and vertical lines is A' ∪ B'.
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'?
Answer
A is the set of triangles in which no triangle is equilateral.
Hence A' is the set of all equilateral triangles.
7. Fill in the blanks to make each of the following a true statement:
(i) A ∪ A' = ..............
(ii) Ф' ∩ A = ..............
(iii) A∩ A' = ..............
(iv) U' ∩ A = ..............
Answer
(i) A ∪ A' = U
(ii) Ф' ∩ A = U∩ A = A
(iii) A ∩ A'= Ф
(iv) U' ∩ A = Ф ∩ A = Ф
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'?
Answer
A is the set of triangles in which no triangle is equilateral.
Hence A' is the set of all equilateral triangles.
7. Fill in the blanks to make each of the following a true statement:
(i) A ∪ A' = ..............
(ii) Ф' ∩ A = ..............
(iii) A∩ A' = ..............
(iv) U' ∩ A = ..............
Answer
(i) A ∪ A' = U
(ii) Ф' ∩ A = U∩ A = A
(iii) A ∩ A'= Ф
(iv) U' ∩ A = Ф ∩ A = Ф