Referring to the AP Inter 1st Year Maths Study Material Chapter 1 Sets Exercise 1e Solutions makes it easier to understand complex problems.
AP Inter 1st Year Maths Sets Solutions Exercise 1e
I.
Question 1.
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:
Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} & B = {2, 3, 5, 7}
i) A ∪ B = {2, 3, 4, 5, 6, 7, 8}
(A ∪ B}’ = U – (A ∪ B) = {1, 9}
A = {1, 3, 5, 7, 9}, B’ = {1, 4, 6, 8, 9}
A’ ∩ B’ = {1, 9}
(A ∪ B)’ = A’ ∩ B’.
ii) A ∩ B = {2}
(A ∩ B)’ = {1, 3, 4, 5, 6, 7, 8, 9}
A’ ∪ B’ = {1, 3, 5, 7,9} ∪ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
(A ∩ B)’ = A’ ∪ B’.
Question 2.
Let U be the set of all triangles in a plane. If A is the set of all triangles with atleast one angle different from 60°, what is A’?
Solution:
Given U be the set of all triangles in a plane.
A is the set of all triangles with atleast one angle different from 60°.
A’ is the set of all equilateral triangles.
Question 3.
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’ ∩ A = U ∩ A = A.
iii) A ∩ A’ = Φ.
iv) U’ ∩ A = Φ ∩ A = Φ.
II.
Question 1.
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 U = {a, b, c, d, e, f, g, h}; A = {a, b, c}; B = {d, e, f, g}; C = {a, c, e, g}; D = {f, g, h, a}
i) A’ = {d, e, f, g, h}.
ii) B’ = {a, b, c, h}.
iii) C’ = {b, d, f, h}.
iv) D’ = {b, c, d, e}.
Question 2.
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)’
ii) A’ ∩ B’
iii) (A ∩ B)’
iv) A’ ∪ B’
III.
Question 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)’
Solution:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6}
i) A’ = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4} = {5, 6, 7, 8, 9}.
ii) B’ = U – B = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8} = {1, 3, 5, 7, 9}.
iii) A ∪ C = {1, 2, 3, 4, 5, 6}
(A ∪ C)’ = U – (A ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6} = {7, 8, 9).
iv) A ∪ B = {1, 2, 3, 4, 6, 8}
(A ∪ B)’ = {5, 7, 9}.
v) (A’)’ = A = {1, 2, 3, 4}.
vi) B – C = {2, 8}
(B – C)’ = {1,3, 4, 5, 6, 7, 9}.
Question 2.
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 U = N = {1, 2, 3, }
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 ∈ N and x is not a multiple of 3}.
iv) {x : x is a prime number}’ = {x : x is a positive composite number and x = 1}.
v) {x : x is a natural number divisible by 3 & 5} = {x : x is a natural number that is not divisible by 3 & 5 }.
vi) {x : x is a perfect square} = {x : x ∈ N and x is not a perfect square}.
vii) {x : x is a perfect cube} = (x : x ∈ N and x is not a perfect cube}.
viii) {x : x + 5 = 8} = {x : x ∈ N and x ≠ 3}.
ix) { x : 2x + 5 = 9} = {x : x ∈ N and x ≠ 2}.
x) {x : x ≥ 7} = {x : x ∈ N and x < 7}.
xi) {x : x ∈ N and 2x + 1 > 10} = {x : x ∈ N and x ≤ 9/2}.