WORKSHEET ON SUBSETS

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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