## Solutions to set-III

In this page, 'Solutions to set-III' we are discussing how to do the problems given in problems on set-III.

### Sub set

Solutions to the problems on subset.

1. Fill in the blanks with ⊆ or ⊈ to make each statement true.

(i) {3} --- {0,2,4,6}

Solution: {3} ⊈ {0,2,4,6}

(ii) {a} ----- {a,b,c}

Solution: {a} ⊆ {a,b,c}

(iii) {8, 18} ---- {18, 8}

Solution:  {8, 18} ⊆ {18, 8}

(iv) {d} ---- {a,b,c}

Solution:  {d} ⊈ {a,b,c}

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?

Solution:  X is not a subset of Y

(ii) Is Y a subset of  X?

solution: Yes, Y is a subset of X.

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, xℕ}

Solution: A is not a subset of B. [Because not all elements of A are elements of B.] 4. Write down the power sets of the following sets.

(i) A = {x, y}

Solution: P(A) = { ∅, {x}, {y}, {x,y}}

(ii) X = {a, b, c}

Solution: P(X) = { {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

(iii) B= {5, 6, 7, 8}

Solution: 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) C =

Solutions: ∅ has no proper subset.

5. Find the number of subsets and the number of proper subsets of the following sets.

(i)    A = { 13, 14, 15, 16, 17, 18}

Solution:  The number of elements in A is 6.

The number of subsets = n[P(A)] = 2⁶ = 64.

The number of proper subsets = 2⁶-1 = 64-1 =63.

(ii)   B = {a, b, c, d, e, f, g}

Solution:  The number of elements in B =7

The number of subsets = n[P(B)] = 2⁷ =128

The number of proper subset =2⁷-1 =128-1 = 127.

(iii)  C = { x: x∈W, xℕ}

Solution:  The number of elements in C = 1

The number of subsets =n[P(C)] =2¹ =2

The number of proper subset = 2-1 =1.

6.(i)If A= ∅, find n[P(A)]

Solution:  n[P(A)]=1 as ∅ itself a subset of ∅.

(ii) If n(A) = 3 find n[P(A)].

Solution: n[P(A)] = 2³ = 8.

(iii) If n[P(A)] =512, find n(A)?

Solution: n[P(A)] =512 = 2

512 = 2⁹ = 2

n = 9

n(A) = 9

(iv) If n[P(A)]=1024, find n(A)?

Solution:    n[P(A)]=1024 = 2

1024  =  2ⁿ   =2¹

n = 10

n(A) = 10

7.7. If n[P(A)] =1, what can you say about the set A?

Solution: A is the empty set.

8.8. 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.

Solution:  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).

Solution:

n(A) = 10

n(B) = 10

n(C) = 11

(iii) State whether the following are True(T) or False (F)

Solution:

(a) 7 ∈ B   - F

(b) 16 ∉ A -T

(c)  {15, 20, 25} ⊂ C - T

(d)  {10, 12} ⊂ B - T

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