Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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If \(A\cap A^{c}\) is asked, what will be its value?
D. \(\varnothing\)
कोई अवयव एक साथ (A) और \(A^{c}\) दोनों में नहीं हो सकता। इसलिए \(A\cap A^{c}=\varnothing\) होता है। / No element can be in both (A) and \(A^{c}\) at the same time. Therefore, \(A\cap A^{c}=\varnothing\).
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If \(A\cup A^{c}\) is asked, what will be its value?
D. (U)
(A) और \(A^{c}\) मिलकर पूरे (U) को बना देते हैं। पूरक गुणों में यह सबसे उपयोगी सर्वसमिका है। / (A) and \(A^{c}\) together form the whole (U). This is one of the most useful complement identities.
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A. ({1,2,3})
\(B^{c}={1,2,3,8,9,10}\), इसलिए \(A\cap B^{c}={1,2,3}\)। पहले पूरक फिर प्रतिच्छेद करें। / \(B^{c}={1,2,3,8,9,10}\), so \(A\cap B^{c}={1,2,3}\). First find the complement and then the intersection.
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If \(U=\{p,q,r,s,t\}\) and \(A=\{p,r,t\}\), which element is in \(A^{c}\)?
C. (q)
(q) (U) में है लेकिन (A) में नहीं है। इसलिए \(q \in A^{c}\) सही है। / (q) is in (U) but not in (A). Therefore, \(q \in A^{c}\) is correct.
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If \(x\in A^{c}\), which statement is correct?
B. \(x\in U\) और \(x\notin A\)\(x\in U\) and \(x\notin A\)
\(x\in A^{c}\) का अर्थ है कि (x) सार्वत्रिक समुच्चय में है पर (A) में नहीं है। प्रतीक पढ़ते समय (U) को न भूलें। / \(x\in A^{c}\) means (x) is in the universal set but not in (A). Do not forget (U) while reading the symbol.
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B. ({1,2,4,5,7,8,10,11})
\(A=\{3,6,9,12\}\), इसलिए पूरक बाकी अवयव हैं। गुणज वाले प्रश्नों में पहले सूची बनाएं। / \(A=\{3,6,9,12\}\), so the complement is the remaining elements. In multiple-based questions, list the set first.
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\(If (U={1,2,3,4,5,6,7,8,9}) and (A={x:x \in U, x\) is prime\(}), what is (A^{c})\)?
B. ({1,4,6,8,9})
\(A=\{2,3,5,7\}\), इसलिए \(A^{c}={1,4,6,8,9}\)। याद रखें कि (1) अभाज्य नहीं है। / \(A=\{2,3,5,7\}\), so \(A^{c}={1,4,6,8,9}\). Remember that (1) is not prime.
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If \(U=\{1,2,3,4,5,6\}\), \(A=\{1,2,3\}\), and \(B=\{3,4,5\}\), what is \(A^{c}\cup B^{c}\)?
A. ({1,2,4,5,6})
\(A^{c}={4,5,6}\) और \(B^{c}={1,2,6}\), उनका संघ ({1,2,4,5,6}) है। यह (\(A\cap B\)^{c}) से भी मिलता है। / \(A^{c}={4,5,6}\) and \(B^{c}={1,2,6}\), so their union is ({1,2,4,5,6}). This also matches (\(A\cap B\)^{c}).
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If \(U=\{1,2,3,4,5,6\}\), \(A=\{1,2,4\}\), and \(B=\{2,4,6\}\), what is \(A^{c}\cap B^{c}\)?
A. ({3,5})
\(A^{c}={3,5,6}\) और \(B^{c}={1,3,5}\), इसलिए प्रतिच्छेद ({3,5}) है। यह (\(A\cup B\)^{c}) जैसा है। / \(A^{c}={3,5,6}\) and \(B^{c}={1,3,5}\), so the intersection is ({3,5}). This is the same as (\(A\cup B\)^{c}).
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If \(A^{c}=\varnothing\), which statement about (A) is correct?
B. (A=U)
यदि पूरक खाली है, तो (A) ने पूरे (U) को कवर कर लिया है। इसलिए (A=U) होगा। / If the complement is empty, then (A) covers the whole (U). Therefore, (A=U).
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