Set Concepts — Exercise 6(C)

Fill in the blanks :

(i) If each element of set P is also an element of set Q, then P is said to be ............ of Q and Q is said to be ................ of P.

(ii) Every set is a ............ of itself.

(iii) The empty set is a ............. of every set.

(iv) If A is proper subset of B, then n(A) ............. n(B).

Answer

(i) If each element of set P is also an element of set Q, then P is said to be subset of Q and Q is said to be superset of P.

(ii) Every set is a subset of itself.

(iii) The empty set is a subset of every set.

(iv) If A is proper subset of B, then n(A) is less than n(B).

If A = {5, 7, 8, 9}, then which of the following are subsets of A ?

(i) B = {5, 8}

(ii) C = {0}

(iii) D = {7, 9, 10}

(iv) E = { }

(v) F = {8, 7, 9, 5}

Answer

(i) B = {5, 8}. Both 5 and 8 are elements of A. B is a subset of A.

(ii) C = {0}. 0 is not an element of A. C is not a subset of A.

(iii) D = {7, 9, 10}. 10 is not an element of A. D is not a subset of A.

(iv) E = { }. The empty set is a subset of every set. E is a subset of A.

(v) F = {8, 7, 9, 5}. All elements 8, 7, 9 and 5 are in A. F is a subset of A.

Hence, (i), (iv) and (v) are subsets of A.

If P = {2, 3, 4, 5}, then which of the following are proper subsets of P ?

(i) A = {3, 4}

(ii) B = { }

(iii) C = {23, 45}

(iv) D = {6, 5, 4}

(v) E = {0}

Answer

A set is a proper subset of P if all its elements are in P and it is not equal to P.

(i) A = {3, 4}. Both 3 and 4 are in P, and A ≠ P. A is a proper subset of P.

(ii) B = { }. The empty set is a proper subset of every non-empty set. B is a proper subset of P.

(iii) C = {23, 45}. 23 and 45 are not in P. C is not a proper subset of P.

(iv) D = {6, 5, 4}. 6 is not in P. D is not a proper subset of P.

(v) E = {0}. 0 is not in P. E is not a proper subset of P.

Hence, A and B are proper subsets of P.

If A = {even numbers less than 12},

B = {2, 4}, C = {1, 2, 3}, D = {2, 6} and E = {4}

State, which of the following statements are true :

(i) B ⊂ A

(ii) C ⊆ A

(iii) D ⊂ C

(iv) D ⊄ A

(v) E ⊇ B

(vi) A ⊇ B ⊇ E

Answer

A = {even numbers less than 12} = {2, 4, 6, 8, 10}, B = {2, 4}, C = {1, 2, 3}, D = {2, 6}, E = {4}.

(i) B ⊂ A. The elements 2 and 4 of B are in A and B ≠ A. True

(ii) C ⊆ A. The elements 1 and 3 of C are not in A. False

(iii) D ⊂ C. The element 6 of D is not in C. False

(iv) D ⊄ A. The elements 2 and 6 of D are both in A, so D ⊂ A. Hence D ⊄ A is incorrect. False

(v) E ⊇ B. This means B ⊆ E. The element 2 of B is not in E = {4}. False

(vi) A ⊇ B ⊇ E. Here B ⊆ A (2, 4 are in A) and E ⊆ B (4 is in B). So A ⊇ B ⊇ E holds. True

Hence, statements (i) and (vi) are true.

Given A = {a, c}, B = {p, q, r} and C = Set of digits used to form the number 1351. Write all the subsets of sets A, B and C.

Answer

A = {a, c}, B = {p, q, r}.

C = Set of digits used to form the number 1351. The digits are 1, 3, 5, 1. Writing each digit only once, C = {1, 3, 5}.

Subsets of A : { }, {a}, {c}, {a, c}

Subsets of B : { }, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}

Subsets of C : { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {1, 5}, {1, 3, 5}

(i) If A = {p, q, r}, then the number of subsets of A = ............

(ii) If B = {5, 4, 6, 8}, then the number of proper subsets of B = ............

(iii) If C = {0}, then the number of subsets of C = ............

(iv) If M = {x : x ∈ N and x < 3}, then M has .............. proper subsets.

Answer

If a set has n elements, then the number of its subsets is 2ⁿ and the number of its proper subsets is 2ⁿ − 1.

(i) A = {p, q, r} has 3 elements, so the number of subsets = 2³ = 8

(ii) B = {5, 4, 6, 8} has 4 elements, so the number of proper subsets = 2⁴ − 1 = 16 − 1 = 15

(iii) C = {0} has 1 element, so the number of subsets = 2¹ = 2

(iv) M = {x : x ∈ N and x < 3} = {1, 2} has 2 elements, so the number of proper subsets = 2² − 1 = 4 − 1 = 3

For the universal set {4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find its subsets A, B, C and D such that :

(i) A = {even numbers}

(ii) B = {odd numbers greater than 8}

(iii) C = {prime numbers}

(iv) D = {even numbers less than 10}.

Answer

The universal set is {4, 5, 6, 7, 8, 9, 10, 11, 12, 13}.

(i) The even numbers in the universal set are 4, 6, 8, 10 and 12.

A = {4, 6, 8, 10, 12}

(ii) The odd numbers greater than 8 in the universal set are 9, 11 and 13.

B = {9, 11, 13}

(iii) The prime numbers in the universal set are 5, 7, 11 and 13.

C = {5, 7, 11, 13}

(iv) The even numbers less than 10 in the universal set are 4, 6 and 8.

D = {4, 6, 8}