The number of subsets of a set containing n elements is (a) \(n\) (b) \(2^n-1\) (c) \(n^2\) (d) \(2^n\) Solution If a set contains \(n\) elements, then each element has two choices: either included in a subset or not included in a subset Therefore, total number of subsets \[ =2 \times 2 \times 2 \cdots […]
The number of subsets of a set containing n elements is(a) n(b) 2ⁿ − 1(c) n²(d) 2ⁿ Read More »
Let A and B be two sets in the same universal set. Then, A − B = (a) \(A \cap B\) (b) \(A’ \cap B\) (c) \(A \cap B’\) (d) none of these Solution By definition, \[ A-B \] means the elements which belong to \(A\) but do not belong to \(B\). Therefore, \[ A-B=A
For any set A, (A′)′ is equal to (a) \(A’\) (b) \(A\) (c) \(\phi\) (d) none of these Solution By the double complement law, \[ (A’)’=A \] Hence, the complement of the complement of a set is the set itself. Answer \[ \boxed{A} \] Correct option: (b) Next Question / Full Exercise
For any set A, (A′)′ is equal to(a) A′(b) A(c) ϕ(d) none of these Read More »
If A and B are two finite sets such that n(A) > n(B) and the difference of the number of elements of the power sets of A and B is 96, then n(A) – n(B) =………….. Solution Let \[ n(A)=m,\qquad n(B)=n \] Number of elements in power sets: \[ n(P(A))=2^m,\qquad n(P(B))=2^n \] Given, \[ 2^m-2^n=96
For any two sets A and B, if n(A) = 15, n(B) = 12, A∩B = Φ and B⊄A, then the maximum and minimum possible values of n(A∆B) are………..and………….respectively. Solution Given, \[ A \cap B = \Phi \] Therefore, \[ A \Delta B = A \cup B \] Hence, \[ n(A \Delta B)=n(A)+n(B) \] \[
A and B are any two non-empty sets and A is proper subset of B. If n(A) = 5, then the minimum possible value of n(A∆ B) is Solution Since \[ A \subset B \] therefore, \[ A \cap B = A \] and \[ A \Delta B = (A-B)\cup(B-A) \] But, \[ A-B=\Phi \]
If n(A∩B) = 5, n(A∩C) = 7 and n(A∩B∩C) = 3, then the minimum possible value of n(B∩C) is…. Solution \[ n(A \cap B)=5 \] \[ n(A \cap C)=7 \] \[ n(A \cap B \cap C)=3 \] Since \[ A \cap B \cap C \subseteq B \cap C \] therefore, \[ n(B \cap C) \ge
If A, B and C are three non-empty sets such that any two of them are disjoint, then \((A \cup B \cup C) \cap (A \cap B \cap C) = …..\) Solution Since any two sets are disjoint, \[ A \cap B = \Phi,\qquad B \cap C = \Phi,\qquad A \cap C = \Phi \]
If n(A∩B) = 10, n(B∩C) = 20 and n(A∩C) = 30, then the greater possible value of n(A ∩ B ∩ C) is….. Solution \[ n(A \cap B)=10 \] \[ n(B \cap C)=20 \] \[ n(A \cap C)=30 \] Since \[ A \cap B \cap C \subseteq A \cap B \] therefore, \[ n(A \cap
Let S = {x : x is a positive multiple of 3 less than 100}, P = {x : x is a prime number less than 20}, Then, n(S) + n(P) = _____ Solution \[ S = \{3,6,9,\ldots,99\} \] Number of multiples of 3 less than 100: \[ \frac{99}{3} = 33 \] Therefore, \[ n(S)=33