MCQ Set Theory

Collection of Multiple choice questions and Answers (MCQs) focuses on “Set Theory” along with their solution and justification.

Let A = \{2, 3, 4\} and X = \{0, 1, 2, 3, 4\}, then which of the following statements is correct?
(A) \{0\} \in A' in X
(B) f \in A' with respect to X
(C) \{0\} \subset A' with respect to X
(D) 0 \subset A' with respect to X.

Answer: (C)
We have, A' in X= The set of elements in X which are not in A = \{0, 1\}
{0} \in A' in X is false, because \{0\} is not an element of A' in X.
f \subset A' in X is false, because f is not an element of A' in X
\{0\} \subset A' in X is correct, because the only element of \{0\} namely 0 also belongs to A' in X.
0 \subset A' in X is false, because 0 is not a set.

If n (U) = 60, n (A) = 35, n (B) = 24 and n (A \cup B)' =10 then n (A \cap B) is
(A) 9
(B) 8
(C) 6
(D) None of these

Answer: (A)
Explanation:
We have,
n (A \cup B) = n (U) - n(A \cup B)' = 60 -10 = 50
Now, n (A \cup B) = n (A) + n(B) - n(A \cap B)
\Rightarrow 50 = 35 + 24 - n (A \cap B)
\Rightarrow n(A \cap B) = 59 - 50 = 9.

Two finite sets have m and n elements, then total number of subsets of the first set is 56 more that the total number of subsets of the second. The values of m and n are,
(A) 7, 6
(B) 6, 3
(D) 5, 1
(D) 8, 7

Solution: (B)
Since the two finite sets have m and n elements, so number of subsets of these sets will be 2^m and 2^n respectively. According to the question
2^m-2^n = 56
putting m = 6, n = 3, we get 2^6 - 2^3 = 56 or 64 - 8 = 56

If A has 3 elements and B has 6 elements, then the minimum number of elements in the set A \cup B is
(A) 6
(B) 3
(C) \phi
(D) None of these

Answer: (A)
Clearly the number of elements in A \cup B will be minimum when A \subset B. Hence the minimum number of elements in A \cup B is the same as the number of elements in B, that is, 6.

Solution: (A)
Clearly the number of elements in A \cup B will be minimum when A \subset B. Hence the minimum number of elements in A \cup B is the same as the number of elements in