Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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If \(U={1,2,3,\ldots,10}\), \(A=\{2,4,6,8,10\}\), and \(C=A^c\), what is \(C^c\cap{1,2,3,4}\)?
A. ({2,4})
\(C=A^c\) है, इसलिए \(C^c=A\) होगा। अब \(A\cap{1,2,3,4}={2,4}\) है। / \(C=A^c\), so \(C^c=A\). Now \(A\cap{1,2,3,4}={2,4}\).
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Which statement proves that (A) and \(A^c\) together form a partition of (U)?
A. \(A\cap A^c=\varnothing\) और \(A\cup A^c=U\)\(A\cap A^c=\varnothing\) and \(A\cup A^c=U\)
विभाजन के लिए भाग असंबद्ध होने चाहिए और उनका संघ पूरा (U) होना चाहिए। (A) और \(A^c\) यही दोनों शर्तें पूरी करते हैं। / For a partition, the parts must be disjoint and their union must be the whole (U). (A) and \(A^c\) satisfy both conditions.
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If \(U\ne\varnothing\), which statement about \(A=A^c\) is correct?
A. ऐसा कोई (A) संभव नहीं हैno such (A) is possible
यदि \(A=A^c\), तो \(A\cap A^c=A\) होगा, पर \(A\cap A^c=\varnothing\) है। \(U\ne\varnothing\) में यह स्थिति संभव नहीं होती। / If \(A=A^c\), then \(A\cap A^c=A\), but \(A\cap A^c=\varnothing\). For \(U\ne\varnothing\), this situation is impossible.
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If (U) is enlarged and (A) remains the same, what may happen to \(A^c\)?
A. \(A^c\) में नए तत्व जुड़ सकते हैंnew elements may be added to \(A^c\)
पूरक (U) पर निर्भर करता है। यदि (U) में नए तत्व आए और वे (A) में न हों, तो वे \(A^c\) में जुड़ेंगे। / Complement depends on (U). If new elements enter (U) and are not in (A), they are added to \(A^c\).
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A. ({6,7})
\(U_1\) में पूरक ({1,3,4}) और \(U_2\) में पूरक ({1,3,4,6,7}) है। अतिरिक्त तत्व ({6,7}) हैं। / Relative to \(U_1\), the complement is ({1,3,4}), and relative to \(U_2\), it is ({1,3,4,6,7}). The extra elements are ({6,7}).
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If \(U={1,2,3,\ldots,12}\), \(A=\{1,2,3,4\}\), and \(B=\{3,4,5,6,7\}\), what is (\(A^c\cap B\)^c)?
A. ({1,2,3,4,8,9,10,11,12})
\(A^c={5,6,7,8,9,10,11,12}\) और \(A^c\cap B={5,6,7}\) है। इसका पूरक (U-{5,6,7}) होगा। / \(A^c={5,6,7,8,9,10,11,12}\) and \(A^c\cap B={5,6,7}\). Its complement is (U-{5,6,7}).
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A. ({6,8,10})
\(A^c={1,3,5,7,9}\) और \(A^c\cup B={1,2,3,4,5,7,9}\) है। (U) में बचे (6,8,10) हैं। / \(A^c={1,3,5,7,9}\), and \(A^c\cup B={1,2,3,4,5,7,9}\). The remaining elements in (U) are (6,8,10).
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A. (16)
(A^c\cap B^c=\(A\cup B\)^c) है। (4) या (6) से विभाज्य संख्याएं (6+4-2=8) हैं, इसलिए पूरक में (24-8=16) तत्व हैं। / (A^c\cap B^c=\(A\cup B\)^c). Numbers divisible by (4) or (6) are (6+4-2=8), so the complement has (24-8=16) elements.
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A. (13)
(A^c\cup B^c=\(A\cap B\)^c) है। \(A\cap B={3,5}\), इसलिए पूरक में (15-2=13) तत्व होंगे। / (A^c\cup B^c=\(A\cap B\)^c). Since \(A\cap B={3,5}\), the complement has (15-2=13) elements.
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If \(U={1,2,3,\ldots,11}\), \(A=\{1,4,7,10\}\), and \(B=\{2,4,6,8,10\}\), what is \(A^c\cap B^c\)?
A. ({3,5,9,11})
(A^c\cap B^c=\(A\cup B\)^c) है। \(A\cup B={1,2,4,6,7,8,10}\), इसलिए बचे तत्व ({3,5,9,11}) हैं। / (A^c\cap B^c=\(A\cup B\)^c). Since \(A\cup B={1,2,4,6,7,8,10}\), the remaining elements are ({3,5,9,11}).
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