यदि \(U=\{1,2,3,4,5,6,7,8,9,10\}\) और \(A=\{2,3,5,7\}\) है, तो (|\mathcal{P}(A')|) कितना होगा?
If \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{2,3,5,7\}\), what is (|\mathcal{P}(A')|)?
#sets
#power-set
#universal-set
#complement
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A (16)
B (32)
C (64)
D (128)
Explanation opens after your attempt
Step 1
Concept
Here (A') has (6) elements, so (|\mathcal{P}(A')|=26 =64). In exams, always find complement with respect to (U).
Step 2
Why this answer is correct
The correct answer is C. (64). Here (A') has (6) elements, so (|\mathcal{P}(A')|=26 =64). In exams, always find complement with respect to (U).
Step 3
Exam Tip
यहां (A') में (6) तत्व हैं, इसलिए (|\mathcal{P}(A')|=26 =64)। परीक्षा में complement हमेशा (U) के सापेक्ष निकालें।
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यदि (|U|=14), \(A\subseteq U\), और (|\mathcal{P}(A)|=64) है, तो (|\mathcal{P}(A')|) का मान क्या है?
If (|U|=14), \(A\subseteq U\), and (|\mathcal{P}(A)|=64), what is the value of (|\mathcal{P}(A')|)?
#sets
#cardinality
#complement
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A \(2^6\)
B \(2^7\)
C \(2^8\)
D \(2^{14}\)
Explanation opens after your attempt
Correct Answer
C. \(2^8\)
Step 1
Concept
Since (|\mathcal{P}(A)|=64=26 ), (|A|=6) and (|A'|=8). Hence (|\mathcal{P}(A')|=28 ).
Step 2
Why this answer is correct
The correct answer is C. \(2^8\). Since (|\mathcal{P}(A)|=64=26 ), (|A|=6) and (|A'|=8). Hence (|\mathcal{P}(A')|=28 ).
Step 3
Exam Tip
(|\mathcal{P}(A)|=64=26 ), इसलिए (|A|=6) और (|A'|=8)। अतः (|\mathcal{P}(A')|=28 )।
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यदि \(A={\varnothing,1,{1}}\) है, तो (\mathcal{P}(A)) में कितने सदस्य होंगे?
If \(A={\varnothing,1,{1}}\), how many members are there in (\mathcal{P}(A))?
#sets
#nested-set
#power-set
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A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The set (A) has (3) distinct elements, so (|\mathcal{P}(A)|=23 =8). In exams, treat \(\varnothing\), (1), and ({1}) as different.
Step 2
Why this answer is correct
The correct answer is C. (8). The set (A) has (3) distinct elements, so (|\mathcal{P}(A)|=23 =8). In exams, treat \(\varnothing\), (1), and ({1}) as different.
Step 3
Exam Tip
समुच्चय (A) में (3) अलग-अलग तत्व हैं, इसलिए (|\mathcal{P}(A)|=23 =8)। परीक्षा में \(\varnothing\), (1), और ({1}) को अलग मानें।
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यदि \(A=\{a,b,c,d\}\) है, तो (\mathcal{P}(A)) में ठीक (2) तत्वों वाले सदस्यों की संख्या कितनी है?
If \(A=\{a,b,c,d\}\), how many members of (\mathcal{P}(A)) have exactly (2) elements?
#sets
#combination
#power-set
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The number of such subsets is \(\binom{4}{2}=6\). In exams, use \(\binom{n}{r}\) for fixed-size subsets.
Step 2
Why this answer is correct
The correct answer is B. (6). The number of such subsets is \(\binom{4}{2}=6\). In exams, use \(\binom{n}{r}\) for fixed-size subsets.
Step 3
Exam Tip
ऐसे subsets की संख्या \(\binom{4}{2}=6\) है। परीक्षा में fixed size subsets के लिए \(\binom{n}{r}\) लगाएं।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,2,3\}\) और \(B=\{3,4,5\}\) है, तो (|\mathcal{P}(\(A\cup B\)')|) कितना है?
If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,2,3\}\), and \(B=\{3,4,5\}\), what is (|\mathcal{P}(\(A\cup B\)')|)?
#sets
#union
#complement
#power-set
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3,4,5}\), so the complement has (3) elements. Hence (|\mathcal{P}(\(A\cup B\)')|=23 =8).
Step 2
Why this answer is correct
The correct answer is C. (8). \(A\cup B={1,2,3,4,5}\), so the complement has (3) elements. Hence (|\mathcal{P}(\(A\cup B\)')|=23 =8).
Step 3
Exam Tip
\(A\cup B={1,2,3,4,5}\), इसलिए complement में (3) तत्व हैं। अतः (|\mathcal{P}(\(A\cup B\)')|=23 =8)।
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यदि (\mathcal{P}(A)=\mathcal{P}(B)) है, तो निश्चित रूप से कौन सा कथन सही है?
If (\mathcal{P}(A)=\mathcal{P}(B)), which statement is definitely true?
#sets
#power-set
#reasoning
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A \(A\subset B\)
B \(B\subset A\)
C (A=B)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Since \(A\in\mathcal{P}(A)\), \(A\in\mathcal{P}(B)\) and \(A\subseteq B\). Similarly \(B\subseteq A\), so (A=B).
Step 2
Why this answer is correct
The correct answer is C. (A=B). Since \(A\in\mathcal{P}(A)\), \(A\in\mathcal{P}(B)\) and \(A\subseteq B\). Similarly \(B\subseteq A\), so (A=B).
Step 3
Exam Tip
क्योंकि \(A\in\mathcal{P}(A)\), इसलिए \(A\in\mathcal{P}(B)\) और \(A\subseteq B\)। इसी तरह \(B\subseteq A\), इसलिए (A=B)।
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यदि \(A\subseteq U\), (|U|=9), और (|A'|=4) है, तो (\mathcal{P}(A)) में कितने proper subsets होंगे?
If \(A\subseteq U\), (|U|=9), and (|A'|=4), how many proper subsets are there in (\mathcal{P}(A))?
#sets
#proper-subset
#universal-set
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A (15)
B (31)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(|A|=9-4=5), so proper subsets of (A) are \(2^5-1=31\). In exams, exclude the whole set for proper subsets.
Step 2
Why this answer is correct
The correct answer is B. (31). (|A|=9-4=5), so proper subsets of (A) are \(2^5-1=31\). In exams, exclude the whole set for proper subsets.
Step 3
Exam Tip
(|A|=9-4=5), इसलिए (A) के proper subsets \(2^5-1=31\) हैं। परीक्षा में proper subset के लिए पूरा set हटाएं।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनमें (1) हो लेकिन (2) न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) contain (1) but not (2)?
#sets
#counting
#power-set
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (2) is excluded, so the remaining (3) elements give \(2^3=8\) subsets. In exams, separate compulsory and forbidden elements.
Step 2
Why this answer is correct
The correct answer is B. (8). (1) is fixed and (2) is excluded, so the remaining (3) elements give \(2^3=8\) subsets. In exams, separate compulsory and forbidden elements.
Step 3
Exam Tip
(1) fixed है और (2) excluded है, इसलिए बाकी (3) तत्वों से \(2^3=8\) subsets बनेंगे। परीक्षा में compulsory और forbidden elements अलग करें।
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यदि \(A=\{p,q,r,s,t,u\}\) है, तो (\mathcal{P}(A)) में (p) और (q) दोनों को न रखने वाले subsets कितने हैं?
If \(A=\{p,q,r,s,t,u\}\), how many subsets in (\mathcal{P}(A)) contain neither (p) nor (q)?
#sets
#counting
#subset
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
After removing (p) and (q), (4) elements remain, so there are \(2^4=16\) subsets. In exams, remove both elements for a neither condition.
Step 2
Why this answer is correct
The correct answer is B. (16). After removing (p) and (q), (4) elements remain, so there are \(2^4=16\) subsets. In exams, remove both elements for a neither condition.
Step 3
Exam Tip
(p) और (q) हटाने पर (4) तत्व बचते हैं, इसलिए \(2^4=16\) subsets होंगे। परीक्षा में neither condition में दोनों तत्व हटाएं।
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यदि (|A|=7) है, तो (\mathcal{P}(A)) में even cardinality वाले subsets की संख्या कितनी है?
If (|A|=7), how many subsets in (\mathcal{P}(A)) have even cardinality?
#sets
#even-subsets
#power-set
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A (32)
B (64)
C (96)
D (128)
Explanation opens after your attempt
Step 1
Concept
When \(|A|=n\geq1\), the number of even cardinality subsets is \(2^{n-1}\). Here it is \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is B. (64). When \(|A|=n\geq1\), the number of even cardinality subsets is \(2^{n-1}\). Here it is \(2^6=64\).
Step 3
Exam Tip
जब \(|A|=n\geq1\), even cardinality subsets की संख्या \(2^{n-1}\) होती है। यहां \(2^6=64\) है।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में odd cardinality और (6) को रखने वाले subsets कितने हैं?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) have odd cardinality and contain (6)?
#sets
#parity
#counting
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(6) is fixed, so an even number must be chosen from the remaining (5) elements. Such choices are \(2^{5-1}=16\).
Step 2
Why this answer is correct
The correct answer is B. (16). (6) is fixed, so an even number must be chosen from the remaining (5) elements. Such choices are \(2^{5-1}=16\).
Step 3
Exam Tip
(6) fixed है, इसलिए बाकी (5) तत्वों में even संख्या चुननी होगी। ऐसे choices \(2^{5-1}=16\) हैं।
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यदि \(A=\{1,{2,3},4\}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का सदस्य है?
If \(A=\{1,{2,3},4\}\), which of the following is a member of (\mathcal{P}(A))?
#sets
#nested-set
#membership
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A ({2,3})
B ({{2,3}})
C ({1,2,3})
D ({{1,4}})
Explanation opens after your attempt
Correct Answer
B. ({{2,3}})
Step 1
Concept
({2,3}) itself is an element of (A), so ({{2,3}}) is a subset of (A). In exams, treat a nested element as one object.
Step 2
Why this answer is correct
The correct answer is B. ({{2,3}}). ({2,3}) itself is an element of (A), so ({{2,3}}) is a subset of (A). In exams, treat a nested element as one object.
Step 3
Exam Tip
({2,3}) स्वयं (A) का तत्व है, इसलिए ({{2,3}}) (A) का subset है। परीक्षा में nested element को एक object मानें।
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यदि \(A=\varnothing\) है, तो (\mathcal{P}(\mathcal{P}(\mathcal{P}(A)))) में कितने सदस्य होंगे?
If \(A=\varnothing\), how many members are there in (\mathcal{P}(\mathcal{P}(\mathcal{P}(A))))?
#sets
#empty-set
#iterated-power-set
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has (1) member and (\mathcal{P}(\mathcal{P}(A))) has (2) members. Hence the next power set has \(2^2=4\) members.
Step 2
Why this answer is correct
The correct answer is B. (4). (\mathcal{P}(A)) has (1) member and (\mathcal{P}(\mathcal{P}(A))) has (2) members. Hence the next power set has \(2^2=4\) members.
Step 3
Exam Tip
(\mathcal{P}(A)) में (1) सदस्य और (\mathcal{P}(\mathcal{P}(A))) में (2) सदस्य हैं। इसलिए अगले power set में \(2^2=4\) सदस्य होंगे।
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यदि (|\mathcal{P}(\mathcal{P}(A))|=256) है, तो (|A|) कितना है?
If (|\mathcal{P}(\mathcal{P}(A))|=256), what is (|A|)?
#sets
#iterated-power-set
#cardinality
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
If (|A|=n), then (|\mathcal{P}(\mathcal{P}(A))|=2^{2^n}). Since \(256=2^8=2^{2^3}\), (n=3).
Step 2
Why this answer is correct
The correct answer is B. (3). If (|A|=n), then (|\mathcal{P}(\mathcal{P}(A))|=2^{2^n}). Since \(256=2^8=2^{2^3}\), (n=3).
Step 3
Exam Tip
यदि (|A|=n), तो (|\mathcal{P}(\mathcal{P}(A))|=2^{2^n})। \(256=2^8=2^{2^3}\), इसलिए (n=3)।
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यदि \(A\subseteq B\), (|A|=3), और (|B|=6) है, तो (|\mathcal{P}(B)-\mathcal{P}(A)|) कितना है?
If \(A\subseteq B\), (|A|=3), and (|B|=6), what is (|\mathcal{P}(B)-\mathcal{P}(A)|)?
#sets
#power-set
#difference
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A (32)
B (48)
C (56)
D (64)
Explanation opens after your attempt
Step 1
Concept
Since \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\), the difference is \(2^6-2^3=64-8=56\). In exams, subtract directly for nested power sets.
Step 2
Why this answer is correct
The correct answer is C. (56). Since \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\), the difference is \(2^6-2^3=64-8=56\). In exams, subtract directly for nested power sets.
Step 3
Exam Tip
क्योंकि \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\), इसलिए अंतर \(2^6-2^3=64-8=56\) है। परीक्षा में nested power sets में सीधे घटाएं।
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यदि \(|A\cap B|=4\) है, तो \(|\mathcal{P}(A)\cap\mathcal{P}(B)|\) कितना होगा?
If \(|A\cap B|=4\), what is \(|\mathcal{P}(A)\cap\mathcal{P}(B)|\)?
#sets
#power-set
#intersection
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}\(A\cap B\)), so the number is \(2^4=16\). In exams, remember the identity for common subsets.
Step 2
Why this answer is correct
The correct answer is C. (16). (\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}\(A\cap B\)), so the number is \(2^4=16\). In exams, remember the identity for common subsets.
Step 3
Exam Tip
(\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}\(A\cap B\)), इसलिए संख्या \(2^4=16\) है। परीक्षा में common subsets की identity याद रखें।
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यदि \(A\cap B=\varnothing\), (|A|=4), और (|B|=3) है, तो \(|\mathcal{P}(A)\cup\mathcal{P}(B)|\) कितना है?
If \(A\cap B=\varnothing\), (|A|=4), and (|B|=3), what is \(|\mathcal{P}(A)\cup\mathcal{P}(B)|\)?
#sets
#disjoint-sets
#union
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A (21)
B (22)
C (23)
D (24)
Explanation opens after your attempt
Step 1
Concept
The only common member of the two power sets is \(\varnothing\). Therefore \(2^4+2^3-1=16+8-1=23\).
Step 2
Why this answer is correct
The correct answer is C. (23). The only common member of the two power sets is \(\varnothing\). Therefore \(2^4+2^3-1=16+8-1=23\).
Step 3
Exam Tip
दोनों power sets में common member केवल \(\varnothing\) है। इसलिए \(2^4+2^3-1=16+8-1=23\)।
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यदि \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,3,5,7,9\}\) और \(B=\{2,3,5,8\}\) है, तो (|\mathcal{P}((A-B)')|) कितना है?
If \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,3,5,7,9\}\), and \(B=\{2,3,5,8\}\), what is (|\mathcal{P}((A-B)')|)?
#sets
#difference
#complement
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A (16)
B (32)
C (64)
D (128)
Explanation opens after your attempt
Step 1
Concept
(A-B={1,7,9}), so its complement has (6) elements. Hence the power set size is \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is C. (64). (A-B={1,7,9}), so its complement has (6) elements. Hence the power set size is \(2^6=64\).
Step 3
Exam Tip
(A-B={1,7,9}), इसलिए उसके complement में (6) तत्व हैं। अतः power set size \(2^6=64\) है।
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यदि \(A=\{1,2,3,4,5,6,7\}\) है, तो (\mathcal{P}(A)) में कम से कम (6) तत्वों वाले subsets कितने हैं?
If \(A=\{1,2,3,4,5,6,7\}\), how many subsets in (\mathcal{P}(A)) have at least (6) elements?
#sets
#combination
#power-set
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A (7)
B (8)
C (14)
D (21)
Explanation opens after your attempt
Step 1
Concept
The number of subsets with at least (6) elements is \(\binom{7}{6}+\binom{7}{7}=7+1=8\). In exams, at least means adding all larger sizes.
Step 2
Why this answer is correct
The correct answer is B. (8). The number of subsets with at least (6) elements is \(\binom{7}{6}+\binom{7}{7}=7+1=8\). In exams, at least means adding all larger sizes.
Step 3
Exam Tip
कम से कम (6) तत्वों वाले subsets की संख्या \(\binom{7}{6}+\binom{7}{7}=7+1=8\) है। परीक्षा में at least का मतलब सभी बड़े sizes जोड़ना है।
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यदि \(A=\{1,2,3,4,5,6,7,8\}\) है, तो (\mathcal{P}(A)) में अधिकतम (2) तत्वों वाले subsets कितने हैं?
If \(A=\{1,2,3,4,5,6,7,8\}\), how many subsets in (\mathcal{P}(A)) have at most (2) elements?
#sets
#combination
#cardinality
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A (29)
B (36)
C (37)
D (64)
Explanation opens after your attempt
Step 1
Concept
The number is \(\binom{8}{0}+\binom{8}{1}+\binom{8}{2}=1+8+28=37\). In exams, include the subset of size (0) in at most.
Step 2
Why this answer is correct
The correct answer is C. (37). The number is \(\binom{8}{0}+\binom{8}{1}+\binom{8}{2}=1+8+28=37\). In exams, include the subset of size (0) in at most.
Step 3
Exam Tip
संख्या \(\binom{8}{0}+\binom{8}{1}+\binom{8}{2}=1+8+28=37\) है। परीक्षा में at most में (0) size वाला subset भी जोड़ें।
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यदि \(A=\{a,b,c,d,e\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनमें (a) और (b) में से ठीक एक हो?
If \(A=\{a,b,c,d,e\}\), how many subsets in (\mathcal{P}(A)) contain exactly one of (a) and (b)?
#sets
#exactly-one
#counting
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A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
There are (2) ways to choose exactly one of (a,b), and the remaining (3) elements are free. Hence \(2\cdot2^3=16\).
Step 2
Why this answer is correct
The correct answer is B. (16). There are (2) ways to choose exactly one of (a,b), and the remaining (3) elements are free. Hence \(2\cdot2^3=16\).
Step 3
Exam Tip
(a,b) में से ठीक एक चुनने के (2) तरीके हैं और बाकी (3) तत्व स्वतंत्र हैं। इसलिए \(2\cdot2^3=16\)।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में (3) तत्वों वाले subsets कितने हैं जो (1) को रखते हैं और (6) को नहीं रखते?
If \(A=\{1,2,3,4,5,6\}\), how many (3)-element subsets in (\mathcal{P}(A)) contain (1) and do not contain (6)?
#sets
#combination
#counting
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A (4)
B (6)
C (10)
D (15)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (6) is excluded, so choose (2) elements from ({2,3,4,5}). The number is \(\binom{4}{2}=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). (1) is fixed and (6) is excluded, so choose (2) elements from ({2,3,4,5}). The number is \(\binom{4}{2}=6\).
Step 3
Exam Tip
(1) fixed और (6) excluded है, इसलिए (2) तत्व ({2,3,4,5}) से चुनेंगे। संख्या \(\binom{4}{2}=6\) है।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जो ({1,2}) से disjoint हैं?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) are disjoint from ({1,2})?
#sets
#disjoint-subsets
#power-set
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
Disjoint subsets are formed only from ({3,4,5}), so there are \(2^3=8\). In exams, remove forbidden elements for a disjoint condition.
Step 2
Why this answer is correct
The correct answer is B. (8). Disjoint subsets are formed only from ({3,4,5}), so there are \(2^3=8\). In exams, remove forbidden elements for a disjoint condition.
Step 3
Exam Tip
Disjoint subsets केवल ({3,4,5}) से बनेंगे, इसलिए \(2^3=8\) हैं। परीक्षा में disjoint condition के लिए forbidden elements हटा दें।
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यदि \(U=\{a,b,c,d,e,f,g,h\}\) और \(A=\{a,b,c\}\) है, तो (\mathcal{P}(U)) के कितने members (A) को subset के रूप में रखते हैं?
If \(U=\{a,b,c,d,e,f,g,h\}\) and \(A=\{a,b,c\}\), how many members of (\mathcal{P}(U)) contain (A) as a subset?
#sets
#universal-set
#counting
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(A) is fixed and the (5) elements of (U-A) are optional. Therefore the number is \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is C. (32). (A) is fixed and the (5) elements of (U-A) are optional. Therefore the number is \(2^5=32\).
Step 3
Exam Tip
(A) fixed है और (U-A) के (5) तत्व optional हैं। इसलिए संख्या \(2^5=32\) है।
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यदि \(U=\{1,2,3,4,5,6,7\}\) और \(A=\{2,4,6\}\) है, तो (\mathcal{P}(U)) के कितने members (A') के subsets हैं?
If \(U=\{1,2,3,4,5,6,7\}\) and \(A=\{2,4,6\}\), how many members of (\mathcal{P}(U)) are subsets of (A')?
#sets
#complement
#universal-set
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(A'={1,3,5,7}) has (4) elements, so its subsets are \(2^4=16\). In exams, this is the cardinality of (\mathcal{P}(A')).
Step 2
Why this answer is correct
The correct answer is B. (16). (A'={1,3,5,7}) has (4) elements, so its subsets are \(2^4=16\). In exams, this is the cardinality of (\mathcal{P}(A')).
Step 3
Exam Tip
(A'={1,3,5,7}) में (4) तत्व हैं, इसलिए उसके subsets \(2^4=16\) हैं। परीक्षा में यह (\mathcal{P}(A')) की cardinality है।
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यदि \(A=\{1,2,3,4\}\) है, तो (\mathcal{P}(A)) के कितने members singleton sets हैं?
If \(A=\{1,2,3,4\}\), how many members of (\mathcal{P}(A)) are singleton sets?
#sets
#singleton
#power-set
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A (1)
B (2)
C (4)
D (16)
Explanation opens after your attempt
Step 1
Concept
The singleton members are ({1},{2},{3},{4}), so the number is (4). In exams, \(\varnothing\) is not a singleton.
Step 2
Why this answer is correct
The correct answer is C. (4). The singleton members are ({1},{2},{3},{4}), so the number is (4). In exams, \(\varnothing\) is not a singleton.
Step 3
Exam Tip
Singleton members ({1},{2},{3},{4}) हैं, इसलिए संख्या (4) है। परीक्षा में \(\varnothing\) singleton नहीं होता।
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यदि (|A|=4) है, तो (\mathcal{P}(\mathcal{P}(A))) में singleton members कितने होंगे?
If (|A|=4), how many singleton members are there in (\mathcal{P}(\mathcal{P}(A)))?
#sets
#iterated-power-set
#singleton
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^4=16\) members, so its power set has (16) singleton members. In exams, the number of singleton members equals the cardinality of the base set.
Step 2
Why this answer is correct
The correct answer is C. (16). (\mathcal{P}(A)) has \(2^4=16\) members, so its power set has (16) singleton members. In exams, the number of singleton members equals the cardinality of the base set.
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^4=16\) सदस्य हैं, इसलिए उसके power set में (16) singleton members होंगे। परीक्षा में singleton members की संख्या base set की cardinality होती है।
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यदि \(A=\{1,2,3\}\) है, तो \({{1},{2}}\subseteq\mathcal{P}(A)\) कथन कैसा है?
If \(A=\{1,2,3\}\), what is the nature of the statement \({{1},{2}}\subseteq\mathcal{P}(A)\)?
#sets
#membership
#subset
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A सत्य / True
B असत्य / False
C निर्धारित नहीं / Not determined
D केवल \(A=\varnothing\) पर सत्य / True only when \(A=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. सत्य / True
Step 1
Concept
Since \({1}\in\mathcal{P}(A)\) and \({2}\in\mathcal{P}(A)\), the given set is a subset of (\mathcal{P}(A)). In exams, read the outer braces carefully.
Step 2
Why this answer is correct
The correct answer is A. सत्य / True. Since \({1}\in\mathcal{P}(A)\) and \({2}\in\mathcal{P}(A)\), the given set is a subset of (\mathcal{P}(A)). In exams, read the outer braces carefully.
Step 3
Exam Tip
क्योंकि \({1}\in\mathcal{P}(A)\) और \({2}\in\mathcal{P}(A)\), इसलिए दिया गया set (\mathcal{P}(A)) का subset है। परीक्षा में outer braces को ध्यान से पढ़ें।
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यदि \(A=\{1,2\}\) है, तो \({1}\in\mathcal{P}(A)\) और \({1}\subseteq\mathcal{P}(A)\) में से कौन सा सही है?
If \(A=\{1,2\}\), which is correct between \({1}\in\mathcal{P}(A)\) and \({1}\subseteq\mathcal{P}(A)\)?
#sets
#membership
#common-mistake
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A केवल \({1}\in\mathcal{P}(A)\)
B केवल \({1}\subseteq\mathcal{P}(A)\)
C दोनों सत्य / Both true
D दोनों असत्य / Both false
Explanation opens after your attempt
Correct Answer
A. केवल \({1}\in\mathcal{P}(A)\)
Step 1
Concept
({1}) is a subset of (A), so \({1}\in\mathcal{P}(A)\) is true. But \({1}\subseteq\mathcal{P}(A)\) would require \(1\in\mathcal{P}(A)\), which is false.
Step 2
Why this answer is correct
The correct answer is A. केवल \({1}\in\mathcal{P}(A)\). ({1}) is a subset of (A), so \({1}\in\mathcal{P}(A)\) is true. But \({1}\subseteq\mathcal{P}(A)\) would require \(1\in\mathcal{P}(A)\), which is false.
Step 3
Exam Tip
({1}) (A) का subset है, इसलिए \({1}\in\mathcal{P}(A)\) सत्य है। पर \({1}\subseteq\mathcal{P}(A)\) के लिए \(1\in\mathcal{P}(A)\) चाहिए, जो गलत है।
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\(यदि (U={x:x\in\mathbb{N},1\leq x\leq 15}) और (A={x:x\in U, x\) is divisible by \(3}) है, तो (|\mathcal{P}(A')|) कितना है\)?
\(If (U={x:x\in\mathbb{N},1\leq x\leq 15}) and (A={x:x\in U, x\) is divisible by \(3}), what is (|\mathcal{P}(A')|)\)?
#sets
#set-builder
#complement
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A \(2^5\)
B \(2^8\)
C \(2^{10}\)
D \(2^{15}\)
Explanation opens after your attempt
Correct Answer
C. \(2^{10}\)
Step 1
Concept
There are (5) numbers divisible by (3) from (1) to (15), so (|A'|=10). Hence (|\mathcal{P}(A')|=2^{10}).
Step 2
Why this answer is correct
The correct answer is C. \(2^{10}\). There are (5) numbers divisible by (3) from (1) to (15), so (|A'|=10). Hence (|\mathcal{P}(A')|=2^{10}).
Step 3
Exam Tip
(1) से (15) तक (3) से विभाज्य (5) संख्याएं हैं, इसलिए (|A'|=10)। अतः (|\mathcal{P}(A')|=2^{10})।
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\(यदि (U={x:x\in\mathbb{N},1\leq x\leq 20}) और (A={x:x\in U, x\) is prime}) है, तो (A') के non-empty subsets कितने हैं?
\(If (U={x:x\in\mathbb{N},1\leq x\leq 20}) and (A={x:x\in U, x\) is prime}), how many non-empty subsets does (A') have?
#sets
#prime-numbers
#complement
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A (2047)
B (4095)
C (8191)
D (16383)
Explanation opens after your attempt
Step 1
Concept
There are (8) primes from (1) to (20), so (|A'|=12). The non-empty subsets are \(2^{12}-1=4095\).
Step 2
Why this answer is correct
The correct answer is B. (4095). There are (8) primes from (1) to (20), so (|A'|=12). The non-empty subsets are \(2^{12}-1=4095\).
Step 3
Exam Tip
(1) से (20) तक primes (8) हैं, इसलिए (|A'|=12)। Non-empty subsets \(2^{12}-1=4095\) हैं।
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यदि \(A={x:x\in\mathbb{Z},-2\leq x\leq 3}\) है, तो (|\mathcal{P}(A)|) कितना है?
If \(A={x:x\in\mathbb{Z},-2\leq x\leq 3}\), what is (|\mathcal{P}(A)|)?
#sets
#set-builder
#cardinality
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A (16)
B (32)
C (64)
D (128)
Explanation opens after your attempt
Step 1
Concept
The set \(A=\{-2,-1,0,1,2,3\}\) has (6) elements. Therefore (|\mathcal{P}(A)|=26 =64).
Step 2
Why this answer is correct
The correct answer is C. (64). The set \(A=\{-2,-1,0,1,2,3\}\) has (6) elements. Therefore (|\mathcal{P}(A)|=26 =64).
Step 3
Exam Tip
समुच्चय \(A=\{-2,-1,0,1,2,3\}\) में (6) तत्व हैं। इसलिए (|\mathcal{P}(A)|=26 =64)।
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\(यदि (U={1,2,3,4,5,6,7,8,9,10,11,12}) और (A={x:x\in U, x\) is odd\(}) है, तो (\mathcal{P}(A')) में (3) तत्वों वाले members कितने हैं\)?
\(If (U={1,2,3,4,5,6,7,8,9,10,11,12}) and (A={x:x\in U, x\) is odd\(}), how many (3)-element members are there in (\mathcal{P}(A'))\)?
#sets
#complement
#combination
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A (10)
B (15)
C (20)
D (30)
Explanation opens after your attempt
Step 1
Concept
(A') contains (6) even numbers, so the number of (3)-element subsets is \(\binom{6}{3}=20\). In exams, first find the size of (A').
Step 2
Why this answer is correct
The correct answer is C. (20). (A') contains (6) even numbers, so the number of (3)-element subsets is \(\binom{6}{3}=20\). In exams, first find the size of (A').
Step 3
Exam Tip
(A') में even numbers (6) हैं, इसलिए (3)-element subsets की संख्या \(\binom{6}{3}=20\) है। परीक्षा में पहले (A') की size निकालें।
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यदि (A) में (n) तत्व हैं और (\mathcal{P}(A)) में (31) proper subsets हैं, तो (n) कितना है?
If (A) has (n) elements and (\mathcal{P}(A)) has (31) proper subsets, what is (n)?
#sets
#proper-subset
#algebra
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A (4)
B (5)
C (6)
D (31)
Explanation opens after your attempt
Step 1
Concept
The number of proper subsets is \(2^n-1=31\). Therefore \(2^n=32\) and (n=5).
Step 2
Why this answer is correct
The correct answer is B. (5). The number of proper subsets is \(2^n-1=31\). Therefore \(2^n=32\) and (n=5).
Step 3
Exam Tip
Proper subsets की संख्या \(2^n-1=31\) है। इसलिए \(2^n=32\) और (n=5)।
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यदि \(A\subseteq U\), (|A|=m), और (|U|=2m) है, तो (|\mathcal{P}(A)|=|\mathcal{P}(A')|) कब होगा?
If \(A\subseteq U\), (|A|=m), and (|U|=2m), when will (|\mathcal{P}(A)|=|\mathcal{P}(A')|)?
#sets
#universal-set
#algebra
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A हमेशा / Always
B कभी नहीं / Never
C केवल (m=0) पर / Only when (m=0)
D केवल (m=1) पर / Only when (m=1)
Explanation opens after your attempt
Correct Answer
A. हमेशा / Always
Step 1
Concept
Since (|A'|=2m-m=m), both power sets have cardinality \(2^m\). In exams, first find the size of the complement.
Step 2
Why this answer is correct
The correct answer is A. हमेशा / Always. Since (|A'|=2m-m=m), both power sets have cardinality \(2^m\). In exams, first find the size of the complement.
Step 3
Exam Tip
क्योंकि (|A'|=2m-m=m), इसलिए दोनों power sets की cardinality \(2^m\) होगी। परीक्षा में complement की size पहले निकालें।
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यदि \(A\subseteq U\), (|U|=18), और (|\mathcal{P}(A')|=1024) है, तो (|\mathcal{P}(A)|) कितना है?
If \(A\subseteq U\), (|U|=18), and (|\mathcal{P}(A')|=1024), what is (|\mathcal{P}(A)|)?
#sets
#cardinality
#universal-set
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A \(2^6\)
B \(2^8\)
C \(2^{10}\)
D \(2^{18}\)
Explanation opens after your attempt
Correct Answer
B. \(2^8\)
Step 1
Concept
(|\mathcal{P}(A')|=1024=2^{10}), so (|A'|=10) and (|A|=8). Hence (|\mathcal{P}(A)|=28 ).
Step 2
Why this answer is correct
The correct answer is B. \(2^8\). (|\mathcal{P}(A')|=1024=2^{10}), so (|A'|=10) and (|A|=8). Hence (|\mathcal{P}(A)|=28 ).
Step 3
Exam Tip
(|\mathcal{P}(A')|=1024=2^{10}), इसलिए (|A'|=10) और (|A|=8)। अतः (|\mathcal{P}(A)|=28 )।
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यदि \(A=\{1,2,3,4\}\) है, तो (\mathcal{P}(A)) में (A) को छोड़कर कितने members होंगे?
If \(A=\{1,2,3,4\}\), how many members are there in (\mathcal{P}(A)) excluding (A) itself?
#sets
#proper-subset
#power-set
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A (14)
B (15)
C (16)
D (17)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^4=16\) members. Excluding (A) itself leaves (16-1=15).
Step 2
Why this answer is correct
The correct answer is B. (15). (\mathcal{P}(A)) has \(2^4=16\) members. Excluding (A) itself leaves (16-1=15).
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^4=16\) members हैं। (A) को हटाने पर (16-1=15) बचेंगे।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनकी cardinality (2) से विभाज्य है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) have cardinality divisible by (2)?
#sets
#even-subsets
#power-set
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A (16)
B (32)
C (48)
D (64)
Explanation opens after your attempt
Step 1
Concept
Cardinality divisible by (2) means even cardinality. A (6)-element set has \(2^{6-1}=32\) even subsets.
Step 2
Why this answer is correct
The correct answer is B. (32). Cardinality divisible by (2) means even cardinality. A (6)-element set has \(2^{6-1}=32\) even subsets.
Step 3
Exam Tip
Cardinality (2) से विभाज्य होने का अर्थ even cardinality है। (6)-element set के even subsets \(2^{6-1}=32\) होते हैं।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जो ({1,2}) को contain करते हैं और (5) को नहीं रखते?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) contain ({1,2}) and do not contain (5)?
#sets
#containment
#counting
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
({1,2}) is fixed and (5) is excluded, so (3,4) are optional. The number is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). ({1,2}) is fixed and (5) is excluded, so (3,4) are optional. The number is \(2^2=4\).
Step 3
Exam Tip
({1,2}) fixed है और (5) excluded है, इसलिए (3,4) optional हैं। संख्या \(2^2=4\) है।
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यदि \(A=\{a,b,c,d,e\}\) है, तो (\mathcal{P}(A)) में (a) या (b) में से कम से कम एक रखने वाले subsets कितने हैं?
If \(A=\{a,b,c,d,e\}\), how many subsets in (\mathcal{P}(A)) contain at least one of (a) or (b)?
#sets
#at-least-one
#counting
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A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^5=32\), and subsets containing neither (a,b) are \(2^3=8\). Therefore the answer is (32-8=24).
Step 2
Why this answer is correct
The correct answer is C. (24). Total subsets are \(2^5=32\), and subsets containing neither (a,b) are \(2^3=8\). Therefore the answer is (32-8=24).
Step 3
Exam Tip
कुल subsets \(2^5=32\) हैं और (a,b) दोनों न रखने वाले \(2^3=8\) हैं। इसलिए उत्तर (32-8=24) है।
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