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 $X.$

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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.

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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

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

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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

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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 $64 - 8 = 56$

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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

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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.

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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

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