Question 1 :
Fill in the blanks with ⊆ or ⊈ to make each statement true.
(i) {3} --- {0, 2, 4, 6}
(ii) {a} ----- {a, b, c}
(iii) {8, 18} ---- {18, 8}
(iv) {d} ---- {a, b, c}
Question 2 :
Let X= {-3, -2,-1, 0, 1, 2} and Y = {x: x is an integer and -3 ≤ x < 2}
(i) Is X a subset of Y ?
(ii) Is Y a subset of X ?
Question 3 :
Examine whether
A = {x: x is a positive integer divisible by 3}
is a subset of
B = {x: x is a multiple of 5, x∈ℕ}
Question 4 :
Write down the power sets of the following sets.
(i) A = {x, y}
(ii) X = {a, b, c}
(iii) B = {5, 6, 7, 8}
(iv) C = ∅
Question 5 :
Find the number of subsets and the number of proper subsets of the following sets.
(i) A = { 13, 14, 15, 16, 17, 18}
(ii) B = {a, b, c, d, e, f, g}
(iii) C = { x: x∈W, x∉ℕ}
Question 6 :
(i) If A= ∅, find n[P(A)]
(ii) If n(A) = 3 find n[P(A)].
(iii) If n[P(A)] = 512, find n(A) ?
(iv) If n[P(A)] = 1024, find n(A)?
Question 7 :
If n[P(A)] = 1, what can you say about the set A?
Question 8 :
Let A = {x: x is a natural number <11}
B = {x: x is an even number 1 < x < 21}
C = {x: x is an integer and 15 ≤ x ≤ 25}
(i) List the elements of A, B, C.
(ii) Find n(A), n(B) and n(C).
(iii) State whether the following are True(T) or False (F)
1. Answer :
(i) {3} ⊈ {0, 2, 4, 6}
(ii) {a} ⊆ {a, b, c}
(iii) {8, 18} ⊆ {18, 8}
(iv) {d} ⊈ {a, b, c}
2. Answer :
(i) X is not a subset of Y
(ii) Yes, Y is a subset of X.
3. Answer :
Let us list out the elements in both sets.
A = {3, 6, 9, 12, 15, ...........}
B = {5, 10, 15, 20, .............}
Every element of set A is not a elements of set B. So, A is not a subset of B.
4. Answer :
(i) P(A) = { ∅, {x}, {y}, {x, y}}
(ii) P(X) = { {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
(iii) P(B) = { {5}, {6}, {7}, {8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8},{7,8}, {5, 6, 7}, {5, 6, 8}, {5, 7, 8}, {6, 7, 8}, {5, 6, 7, 8}}
(iv) ∅ has no proper subset.
5. Answer :
The number of elements in A is 6.
The number of subsets :
n[P(A)] = 2n
n[P(A)] = 26
n[P(A)] = 64
The number of proper subsets :
= 2n-1
= 64-1
= 63
So, number of subsets = 64 and number of proper subsets = 63.
(ii) B = {a, b, c, d, e, f, g}
n(B) = 7
Number of subsets :
n[P(B)] = 2⁷
n[P(B)] = 128
Number of proper subsets :
= 27-1
= 128-1
= 127
So, number of subsets = 128 and number of proper subsets = 127.
(iii) C = { x: x∈W, x∉ℕ}
n(C) = 1
Number of subsets :
n[P(C)] = 21
n[P(C)] = 2
Number of proper subsets :
= 21-1
= 2-1
= 1
So, number of subsets = 2 and number of proper subsets = 1.
6. Answer :
(i) n[P(A)] = 1 as ∅ itself a subset of ∅.
(ii) n[P(A)] = 23 = 8
(iii) n[P(A)] = 512 = 2ⁿ
512 = 29
n = 9
n(A) = 9
(iv) If n[P(A)] = 1024, find n(A)?
n[P(A)] = 1024 = 2ⁿ
1024 = 2ⁿ
210 = 2ⁿ
n = 10
n(A) = 10
7. Answer :
A is the empty set.
8. Answer :
Let A = {x: x is a natural number <11}
B = {x: x is an even number 1 < x < 21}
C = {x: x is an integer and 15 ≤ x ≤ 25}
(i) List the elements of A, B, C.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
C = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}
(ii) Find n(A), n(B) and n(C).
n(A) = 10
n(B) = 10
n(C) = 11
(iii) State whether the following are True(T) or False (F)
(a) 7 ∈ B - F
(b) 16 ∉ A -T
(c) {15, 20, 25} ⊂ C - T
(d) {10, 12} ⊂ B - T
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