CBSE Class 10 Mathematics Sets MCQs Set 16

Answer: (a) {1, 2, 3, 4, 5, 6}

Explanation:
1. \( N \) is the set of natural numbers 1, 2, 3, ..., so \( x^{2} \le 36 \) gives \( x = 1, 2, 3, 4, 5, 6 \).

Teacher's Note:
1. Natural numbers do not include 0 or negatives.

Answer: (b) 1

Explanation:
1. The empty set has exactly one subset, the empty set itself \( (2^{0} = 1) \).

Teacher's Note:
1. A set with \( n \) elements has 2ⁿ subsets.

Answer: (d) The set of odd natural numbers and the set of even natural numbers

Explanation:
1. 21 is divisible by both 3 and 7, and 15 is a multiple of both 3 and 5, so \( (b) \) and \( (c) \) share elements; the natural numbers are inside the integers.
2. No natural number is both odd and even, so those sets are disjoint.

Teacher's Note:
1. Disjoint sets have no common element.

Answer: (c) 8

Explanation:
1. \( 2^{3} = 8 \) subsets.

Teacher's Note:
1. Count 2ⁿ for \( n \) elements.

Answer: (b) Two sets are disjoint if there are no common elements in both the sets

Explanation:
1. Disjoint means no common element. The other statements are false: equal sets have the same elements, integers include negatives and zero, and the empty set is finite.

Teacher's Note:
1. Recall each definition carefully.

Answer: (d) 4

Explanation:
1. \( P(A) \) = {∅} has 1 element. \( P(P(A)) \) has \( 2^{1} = 2 \) elements. \( P(P(P(A))) \) has \( 2^{2} = 4 \) elements.

Teacher's Note:
1. Each application of the power set takes 2 to the number of elements.

Answer: (a) 8

Explanation:
1. A and \( N \) are forced in. Each of the other 3 elements may be included or not: \( 2^{3} = 8 \).

Teacher's Note:
1. Fix the required elements and count the rest.

Answer: (d) 7

Explanation:
1. A has 3 elements (the third is the set {3, 4}), so it has \( 2^{3} = 8 \) subsets and \( 8 - 1 = 7 \) proper subsets.

Teacher's Note:
1. A proper subset excludes the set itself.

Answer: (c) \( 2^{2^{n}} \)

Explanation:
1. \( P(A) \) has 2ⁿ elements, so \( P(P(A)) \) has \( 2^{2^{n}} \) elements.

Teacher's Note:
1. Apply the 2^(number of elements) rule twice.

Answer: (d) {5} ∈ A

Explanation:
1. The elements of \( A \) are 1, 2, {3, 4} and {5}, so \( n(A) = 4 \).
2. {5} is an element of \( A \), so {5} ∈ \( A \) is true; 3 is not an element and {3, 4} is an element, not a subset.

Teacher's Note:
1. Distinguish between an element and a subset.

Answer: (d) {φ, {φ}, {1, φ}} ⊂ A

Explanation:
1. φ, {φ} and {1, φ} are all elements of \( A \), so the set of those three is a subset of \( A \).
2. {1} is not an element, 2 is not a set, and {2}, {1} are not elements.

Teacher's Note:
1. A set is a subset when every one of its members belongs to \( A \).

Answer: (a) A ⊂ B

Explanation:
1. A ∩ \( B = A \) means every element of \( A \) is in \( B \), so \( A \) ⊂ \( B \).

Teacher's Note:
1. A ∩ \( B \) is contained in both sets.

Answer: (a) A = B

Explanation:
1. A ∪ \( B \) contains \( A \) ∩ \( B \), with equality only when every element of either set is in both, i.e. \( A = B \).

Teacher's Note:
1. Try \( A = B \) as a test case.

Answer: (b) A ∩ B′

Explanation:
1. A \( - B \) is the set of elements of \( A \) that are not in \( B \), i.e. \( A \) ∩ B′.

Teacher's Note:
1. Remember the standard identity.

Answer: (d) {6}

Explanation:
1. A′ = {6, 7}. Removing the elements of B (7 is in \( B \)) leaves {6}.

Teacher's Note:
1. Find the complement first.

Answer: (b) {1, 6, 9}

Explanation:
1. Elements of \( A \) not in \( B \): 1, 6 and 9.

Teacher's Note:
1. Check each element of \( A \) against \( B \).

Answer: (a) A ∪ φ = A

Explanation:
1. The union with the empty set adds nothing, so \( A \) ∪ \( \phi = A \).

Teacher's Note:
1. Know the identity laws.

Answer: (b) the intersection of the two sets

Explanation:
1. Common elements make up the intersection.

Teacher's Note:
1. Union means all elements, intersection means common ones.

Answer: (a) 175

Explanation:
1. Everyone speaks at least one language: \( 200 + 125 \) + both = 500, so both = 175.

Teacher's Note:
1. Use n(A ∪ \( B \)) = (only \( A \)) + (only \( B \)) + (both).

Answer: (c) 2300

Explanation:
1. Readers of at least one paper = \( 3500 + 3000 - 800 = 5700 \).
2. Neither = \( 8000 - 5700 = 2300 \).

Teacher's Note:
1. Subtract the union from the total.

Answer: (a) 373

Explanation:
1. n(A ∩ B) = n(A) − n(A − B) = 115 − 47 = 68.
2. n(A ∪ B) = 115 + 326 − 68 = 373.

Teacher's Note:
1. Find the intersection first.

Answer: (c) 4

Explanation:
1. Only mathematics = \( 15 - (5 + 9 - 3) = 4 \). The 3 in all three subjects are counted in both pair overlaps, hence the \( + 3 \).

Teacher's Note:
1. Draw a Venn diagram with three overlapping circles and fill the innermost region first (all three subjects), then the regions for exactly two subjects, then exactly one.