Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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Showing 481-490 of 509 questions.
If \(A\cap B=B\), which statement is correct?
A. \(B\subseteq A\)
साझा भाग पूरा (B) है, इसलिए (B) का हर सदस्य (A) में है। यही \(B\subseteq A\) है। / The common part is the whole of (B), so every element of (B) is in (A). This means \(B\subseteq A\).
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If (A-B=A), what conclusion about (A) and (B) is correct?
A. \(A\cap B=\emptyset\)
(A) से (B) हटाने पर (A) ही बचा, यानी कोई साझा सदस्य नहीं था। इसलिए दोनों असंबद्ध हैं। / Removing (B) from (A) still leaves (A), so there was no common element. Hence they are disjoint.
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If \(A\subseteq B\), what is the correct relation between (\mathcal{P}(A)) and (\mathcal{P}(B))?
A. \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\)
(A) का हर उपसमुच्चय (B) का भी उपसमुच्चय होगा। इसलिए घात समुच्चय में भी यही दिशा रहती है। / Every subset of (A) is also a subset of (B). So the same direction holds for power sets.
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If \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\), which conclusion is correct?
A. \(A\subseteq B\)
(A) स्वयं (\mathcal{P}(A)) का सदस्य है, इसलिए (A) (\mathcal{P}(B)) में भी होगा। इसका अर्थ \(A\subseteq B\) है। / The set (A) itself is an element of (\mathcal{P}(A)), so it is also in (\mathcal{P}(B)). That means \(A\subseteq B\).
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B. (4)
(\mathcal{P}(B)) में (8) और (\mathcal{P}(A)) में (4) सदस्य हैं। अंतर में (8-4=4) सदस्य होंगे। / (\mathcal{P}(B)) has (8) elements and (\mathcal{P}(A)) has (4). The difference contains (8-4=4) elements.
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If \(A=\{1,2,3,4\}\), how many subsets can be formed using only even elements?
C. (4)
सम सदस्य (2,4) हैं। इनके सभी उपसमुच्चयों की संख्या \(2^2=4\) है। / The even elements are (2,4). The number of their subsets is \(2^2=4\).
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If \(A=\{1,2,3,4,5,6\}\), how many subsets have exactly two odd elements?
D. (24)
तीन विषम संख्याओं में से दो चुनें और तीन सम संख्याएँ वैकल्पिक हैं। संख्या \(\binom{3}{2}\times2^3=24\) है। / Choose two from the three odd numbers, and the three even numbers are optional. The count is \(\binom{3}{2}\times2^3=24\).
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If \(A=\{1,2,3,4,5\}\), how many subsets have at least (4) elements?
B. (6)
चार-सदस्यीय उपसमुच्चय \(\binom{5}{4}=5\) और पांच-सदस्यीय (1) है। कुल (5+1=6) है। / There are \(\binom{5}{4}=5\) four-element subsets and (1) five-element subset. Total is (5+1=6).
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If \(A=\{a,b,c,d,e\}\), how many proper subsets do not contain (a)?
C. (16)
(a) को हटाकर (4) सदस्य बचते हैं, जिनके \(2^4=16\) उपसमुच्चय हैं। ये सभी (A) के उचित उपसमुच्चय हैं। / After excluding (a), (4) elements remain, giving \(2^4=16\) subsets. All of them are proper subsets of (A).
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If \(A=\{1,2,3,4\}\), how many subsets either contain both (2,3) or contain neither of them?
C. (8)
(2,3) के लिए दो वैध स्थितियां हैं और (1,4) स्वतंत्र हैं। इसलिए \(2\times2^2=8\) है। / There are two valid states for (2,3), and (1,4) are free. Hence \(2\times2^2=8\).
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