यदि \(A=\{0,2,4,6\}\) है, तो (\mathcal{P}(A)) में कुल कितने तत्व होंगे?
If \(A=\{0,2,4,6\}\), how many elements will (\mathcal{P}(A)) have?
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A (8)
B (12)
C (16)
D (24)
Explanation opens after your attempt
Step 1
Concept
Set (A) has (4) elements so (\mathcal{P}(A)) has \(2^4=16\) elements. In exams first count the distinct elements of the original set.
Step 2
Why this answer is correct
The correct answer is C. (16). Set (A) has (4) elements so (\mathcal{P}(A)) has \(2^4=16\) elements. In exams first count the distinct elements of the original set.
Step 3
Exam Tip
(A) में (4) तत्व हैं इसलिए (\mathcal{P}(A)) में \(2^4=16\) तत्व होंगे। परीक्षा में पहले मूल समुच्चय के अलग-अलग तत्व गिनें।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\) और \(A=\{2,4,6,8\}\) है, तो (n(\mathcal{P}(A'))) कितना होगा?
If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{2,4,6,8\}\), what is (n(\mathcal{P}(A')))?
#sets
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#universal set
#complement
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(A'={1,3,5,7}), so (n(A')=4). Hence (n(\mathcal{P}(A'))=24 =16).
Step 2
Why this answer is correct
The correct answer is C. (16). (A'={1,3,5,7}), so (n(A')=4). Hence (n(\mathcal{P}(A'))=24 =16).
Step 3
Exam Tip
(A'={1,3,5,7}) है, इसलिए (n(A')=4)। अतः (n(\mathcal{P}(A'))=24 =16) होगा।
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यदि (n(\mathcal{P}(A))=512) है, तो (n(A)) का मान क्या होगा?
If (n(\mathcal{P}(A))=512), what is the value of (n(A))?
#sets
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#elements
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
\(512=2^9\) so (n(A)=9). Always use the \(2^n\) rule for power set questions.
Step 2
Why this answer is correct
The correct answer is C. (9). \(512=2^9\) so (n(A)=9). Always use the \(2^n\) rule for power set questions.
Step 3
Exam Tip
\(512=2^9\) इसलिए (n(A)=9) होगा। घात समुच्चय में हमेशा \(2^n\) नियम लगाएं।
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यदि किसी समुच्चय (B) के कुल उपसमुच्चय (1024) हैं, तो उसके उचित उपसमुच्चयों की संख्या कितनी होगी?
If a set (B) has (1024) total subsets, how many proper subsets does it have?
#sets
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#power set
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A (1023)
B (1024)
C (512)
D (10)
Explanation opens after your attempt
Step 1
Concept
A proper subset does not include the whole set itself. Therefore the number is (1024-1=1023).
Step 2
Why this answer is correct
The correct answer is A. (1023). A proper subset does not include the whole set itself. Therefore the number is (1024-1=1023).
Step 3
Exam Tip
उचित उपसमुच्चय में पूरा समुच्चय स्वयं शामिल नहीं होता। इसलिए संख्या (1024-1=1023) है।
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यदि \(A=\{r,s,t,u\}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व है?
If \(A=\{r,s,t,u\}\), which of the following is an element of (\mathcal{P}(A))?
#sets
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#subset
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A (r)
B ({r,t})
C ({r,v})
D ({s,t,v})
Explanation opens after your attempt
Correct Answer
B. ({r,t})
Step 1
Concept
Every element of a power set is a subset of the original set. ({r,t}) contains only elements of (A) so it is correct.
Step 2
Why this answer is correct
The correct answer is B. ({r,t}). Every element of a power set is a subset of the original set. ({r,t}) contains only elements of (A) so it is correct.
Step 3
Exam Tip
घात समुच्चय का प्रत्येक तत्व मूल समुच्चय का उपसमुच्चय होता है। ({r,t}) में केवल (A) के तत्व हैं इसलिए यह सही है।
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यदि \(A=\{0,{1},2\}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व नहीं है?
If \(A=\{0,{1},2\}\), which of the following is not an element of (\mathcal{P}(A))?
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#nested set
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A ({0,2})
B ({{1}})
C ({0,{1}})
D ({0,1})
Explanation opens after your attempt
Correct Answer
D. ({0,1})
Step 1
Concept
(1) itself is not an element of (A); ({1}) is an element. So ({0,1}) is not a subset of (A).
Step 2
Why this answer is correct
The correct answer is D. ({0,1}). (1) itself is not an element of (A); ({1}) is an element. So ({0,1}) is not a subset of (A).
Step 3
Exam Tip
(1) स्वयं (A) का तत्व नहीं है बल्कि ({1}) तत्व है। इसलिए ({0,1}), (A) का उपसमुच्चय नहीं है।
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यदि \(A={\varnothing,{\varnothing}}\) है, तो (n(\mathcal{P}(A))) कितना होगा?
If \(A={\varnothing,{\varnothing}}\), what is (n(\mathcal{P}(A)))?
#sets
#empty set
#power set
#class 11
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
Set (A) has two distinct elements so its power set has \(2^2=4\) elements. Treat \(\varnothing\) and \({\varnothing}\) as different.
Step 2
Why this answer is correct
The correct answer is C. (4). Set (A) has two distinct elements so its power set has \(2^2=4\) elements. Treat \(\varnothing\) and \({\varnothing}\) as different.
Step 3
Exam Tip
(A) में दो अलग-अलग तत्व हैं इसलिए उसके घात समुच्चय में \(2^2=4\) तत्व होंगे। \(\varnothing\) और \({\varnothing}\) को अलग समझें।
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(\mathcal{P}(\mathcal{P}({a}))) में कितने तत्व होंगे?
How many elements are there in (\mathcal{P}(\mathcal{P}({a})))?
#sets
#iterated power set
#singleton
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}({a})) has (2) elements. Therefore its power set has \(2^2=4\) elements.
Step 2
Why this answer is correct
The correct answer is B. (4). (\mathcal{P}({a})) has (2) elements. Therefore its power set has \(2^2=4\) elements.
Step 3
Exam Tip
(\mathcal{P}({a})) में (2) तत्व होते हैं। इसलिए उसके घात समुच्चय में \(2^2=4\) तत्व होंगे।
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यदि \(U=\{a,b,c,d,e,f\}\) और \(A=\{a,c,f\}\) है, तो (A') क्या होगा?
If \(U=\{a,b,c,d,e,f\}\) and \(A=\{a,c,f\}\), what is (A')?
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A ({b,d,e})
B ({a,c,f})
C ({a,b,c})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({b,d,e})
Step 1
Concept
(A') contains elements of (U) that are not in (A). Here those elements are (b,d,e).
Step 2
Why this answer is correct
The correct answer is A. ({b,d,e}). (A') contains elements of (U) that are not in (A). Here those elements are (b,d,e).
Step 3
Exam Tip
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। यहां वे तत्व (b,d,e) हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10\}\) और \(A=\{1,4,9\}\) है, तो (A') में कितने तत्व होंगे?
If \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{1,4,9\}\), how many elements will (A') have?
#sets
#complement
#cardinality
#class 11
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A (3)
B (6)
C (7)
D (10)
Explanation opens after your attempt
Step 1
Concept
(U) has (10) elements and (A) has (3) elements. Hence (n(A')=10-3=7).
Step 2
Why this answer is correct
The correct answer is C. (7). (U) has (10) elements and (A) has (3) elements. Hence (n(A')=10-3=7).
Step 3
Exam Tip
(U) में (10) और (A) में (3) तत्व हैं। अतः (n(A')=10-3=7)।
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एक विद्यालय के (75) विद्यार्थियों में (32) संगीत में और (28) नाटक में भाग लेते हैं। यदि (10) विद्यार्थी दोनों में भाग लेते हैं, तो किसी भी गतिविधि में भाग न लेने वाले विद्यार्थी कितने हैं?
In a school of (75) students, (32) take part in music and (28) take part in drama. If (10) students take part in both, how many take part in neither activity?
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A (20)
B (25)
C (30)
D (35)
Explanation opens after your attempt
Step 1
Concept
(n\(A\cup B\)=32+28-10=50). Students in neither activity are (75-50=25).
Step 2
Why this answer is correct
The correct answer is B. (25). (n\(A\cup B\)=32+28-10=50). Students in neither activity are (75-50=25).
Step 3
Exam Tip
(n\(A\cup B\)=32+28-10=50) होगा। किसी में नहीं (75-50=25) विद्यार्थी हैं।
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यदि (A) और (B), (U) के उपसमुच्चय हैं, तो (\(A\cap B\)') किसके बराबर होता है?
If (A) and (B) are subsets of (U), what is (\(A\cap B\)') equal to?
#sets
#de morgan law
#complement
#class 11
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A \(A'\cap B'\)
B \(A'\cup B'\)
C \(A\cup B\)
D \(A\cap B\)
Explanation opens after your attempt
Correct Answer
B. \(A'\cup B'\)
Step 1
Concept
By De Morgan's law, (\(A\cap B\)'=A'\cup B'). The complement of an intersection includes elements outside at least one set.
Step 2
Why this answer is correct
The correct answer is B. \(A'\cup B'\). By De Morgan's law, (\(A\cap B\)'=A'\cup B'). The complement of an intersection includes elements outside at least one set.
Step 3
Exam Tip
डी मॉर्गन नियम के अनुसार (\(A\cap B\)'=A'\cup B')। प्रतिच्छेद के पूरक में कम से कम एक समुच्चय से बाहर वाले तत्व आते हैं।
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यदि \(A\subseteq U\) और \(A'= \varnothing\) है, तो कौन सा निष्कर्ष सही है?
If \(A\subseteq U\) and \(A'= \varnothing\), which conclusion is correct?
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#universal set
#complement reasoning
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A \(A=\varnothing\)
B (A=U)
C \(U=\varnothing\)
D \(A\cap U=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
If the complement is empty, no element of (U) lies outside (A). Therefore (A=U).
Step 2
Why this answer is correct
The correct answer is B. (A=U). If the complement is empty, no element of (U) lies outside (A). Therefore (A=U).
Step 3
Exam Tip
पूरक खाली है तो (U) में (A) के बाहर कोई तत्व नहीं है। इसलिए (A=U) होगा।
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यदि (n(U)=80) और (n(A')=19) है, तो (n(A)) कितना होगा?
If (n(U)=80) and (n(A')=19), what is (n(A))?
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#complement
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#class 11
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A (19)
B (41)
C (61)
D (99)
Explanation opens after your attempt
Step 1
Concept
(A) and (A') together form the whole (U). So (n(A)=80-19=61).
Step 2
Why this answer is correct
The correct answer is C. (61). (A) and (A') together form the whole (U). So (n(A)=80-19=61).
Step 3
Exam Tip
(A) और (A') मिलकर पूरा (U) बनाते हैं। इसलिए (n(A)=80-19=61)।
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यदि \(A={x:x\) संख्या (24) का धनात्मक भाजक है(}), तो (n(\mathcal{P}(A))) क्या होगा?
If \(A={x:x\) is a positive divisor of (24)(}), what is (n(\mathcal{P}(A)))?
#sets
#power set
#divisors
#class 11
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A (64)
B (128)
C (256)
D (24)
Explanation opens after your attempt
Step 1
Concept
The divisors of (24) are (1,2,3,4,6,8,12,24), so (n(A)=8). Hence (n(\mathcal{P}(A))=28 =256).
Step 2
Why this answer is correct
The correct answer is C. (256). The divisors of (24) are (1,2,3,4,6,8,12,24), so (n(A)=8). Hence (n(\mathcal{P}(A))=28 =256).
Step 3
Exam Tip
(24) के भाजक (1,2,3,4,6,8,12,24) हैं इसलिए (n(A)=8)। अतः (n(\mathcal{P}(A))=28 =256)।
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यदि \(A=\{k\}\) है, तो (\mathcal{P}(A)) कौन सा है?
If \(A=\{k\}\), which is (\mathcal{P}(A))?
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#singleton
#power set
#class 11
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A ({k})
B \({\varnothing,k}\)
C \({\varnothing,{k}}\)
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
C. \({\varnothing,{k}}\)
Step 1
Concept
The subsets of a singleton set are \(\varnothing\) and the set itself. So the power set contains ({k}), not just (k).
Step 2
Why this answer is correct
The correct answer is C. \({\varnothing,{k}}\). The subsets of a singleton set are \(\varnothing\) and the set itself. So the power set contains ({k}), not just (k).
Step 3
Exam Tip
एकल समुच्चय के उपसमुच्चय \(\varnothing\) और वही समुच्चय होते हैं। इसलिए घात समुच्चय में ({k}) आएगा, केवल (k) नहीं।
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यदि \({m,n}\in \mathcal{P}(A)\) है, तो कौन सा कथन निश्चित रूप से सही है?
If \({m,n}\in \mathcal{P}(A)\), which statement is definitely true?
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#element subset
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A \(m\in A\) और \(n\in A\) / \(m\in A\) and \(n\in A\)
B \({m,n}\not\subseteq A\)
C \(A=\{m,n\}\)
D \(m\notin A\)
Explanation opens after your attempt
Correct Answer
A. \(m\in A\) और \(n\in A\) / \(m\in A\) and \(n\in A\)
Step 1
Concept
\({m,n}\in \mathcal{P}(A)\) means \({m,n}\subseteq A\). Therefore both (m) and (n) are elements of (A).
Step 2
Why this answer is correct
The correct answer is A. \(m\in A\) और \(n\in A\) / \(m\in A\) and \(n\in A\). \({m,n}\in \mathcal{P}(A)\) means \({m,n}\subseteq A\). Therefore both (m) and (n) are elements of (A).
Step 3
Exam Tip
\({m,n}\in \mathcal{P}(A)\) का अर्थ है \({m,n}\subseteq A\)। इसलिए (m) और (n) दोनों (A) के तत्व होंगे।
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यदि \(A={\varnothing,{\varnothing}}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व है?
If \(A={\varnothing,{\varnothing}}\), which of the following is an element of (\mathcal{P}(A))?
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#empty set
#power set
#class 11
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A \({\varnothing}\)
B \({{\varnothing}}\)
C \(\varnothing\)
D उपरोक्त सभी / All of these
Explanation opens after your attempt
Correct Answer
D. उपरोक्त सभी / All of these
Step 1
Concept
All three given options are subsets of (A). Therefore all of them are elements of (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is D. उपरोक्त सभी / All of these. All three given options are subsets of (A). Therefore all of them are elements of (\mathcal{P}(A)).
Step 3
Exam Tip
दिए गए तीनों विकल्प (A) के उपसमुच्चय हैं। इसलिए ये सभी (\mathcal{P}(A)) के तत्व हैं।
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किसी समुच्चय (A) के लिए कौन सा कथन हमेशा सत्य है?
Which statement is always true for a set (A)?
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#empty set
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A \(A\in A\)
B \(\varnothing\in A\)
C \(\varnothing\in \mathcal{P}(A)\)
D \(U\in \mathcal{P}(A)\)
Explanation opens after your attempt
Correct Answer
C. \(\varnothing\in \mathcal{P}(A)\)
Step 1
Concept
\(\varnothing\) is a subset of every set. Therefore \(\varnothing\) is always an element of (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is C. \(\varnothing\in \mathcal{P}(A)\). \(\varnothing\) is a subset of every set. Therefore \(\varnothing\) is always an element of (\mathcal{P}(A)).
Step 3
Exam Tip
\(\varnothing\) हर समुच्चय का उपसमुच्चय है। इसलिए \(\varnothing\) हमेशा (\mathcal{P}(A)) का तत्व होगा।
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यदि (A) में (4) तत्व हैं, तो (A) के अरिक्त उचित उपसमुच्चयों की संख्या कितनी होगी?
If (A) has (4) elements, how many non-empty proper subsets does (A) have?
#sets
#proper subset
#non empty
#class 11
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A (14)
B (15)
C (16)
D (12)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\). Removing \(\varnothing\) and (A) leaves (14) non-empty proper subsets.
Step 2
Why this answer is correct
The correct answer is A. (14). Total subsets are \(2^4=16\). Removing \(\varnothing\) and (A) leaves (14) non-empty proper subsets.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं। \(\varnothing\) और (A) को हटाने पर (14) अरिक्त उचित उपसमुच्चय बचते हैं।
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यदि \(A=\{2,4,6,8,10\}\) है, तो (\mathcal{P}(A)) में ठीक (2) तत्व वाले समुच्चयों की संख्या कितनी है?
If \(A=\{2,4,6,8,10\}\), how many sets in (\mathcal{P}(A)) have exactly (2) elements?
#sets
#power set
#combination
#class 11
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A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly (2) elements is \(\binom{5}{2}=10\). Each choice becomes an element of the power set.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose exactly (2) elements is \(\binom{5}{2}=10\). Each choice becomes an element of the power set.
Step 3
Exam Tip
ठीक (2) तत्व चुनने की संख्या \(\binom{5}{2}=10\) है। हर चयन घात समुच्चय का एक तत्व बनता है।
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यदि (A) में (7) तत्व हैं, तो (A) के ऐसे उपसमुच्चय कितने हैं जिनमें एक निश्चित तत्व अवश्य हो?
If (A) has (7) elements, how many subsets of (A) must contain one fixed element?
#sets
#power set
#subset counting
#class 11
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A (32)
B (64)
C (96)
D (128)
Explanation opens after your attempt
Step 1
Concept
One element is fixed to be included and the remaining (6) elements are free. So the number of subsets is \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is B. (64). One element is fixed to be included and the remaining (6) elements are free. So the number of subsets is \(2^6=64\).
Step 3
Exam Tip
एक तत्व को रखना तय है और बाकी (6) तत्व स्वतंत्र हैं। इसलिए उपसमुच्चयों की संख्या \(2^6=64\) होगी।
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यदि (A) में (5) तत्व हैं, तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी होगी जिनमें दो निश्चित तत्व न हों?
If (A) has (5) elements, how many subsets of (A) do not contain two fixed elements?
#sets
#power set
#restricted subsets
#class 11
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
After excluding two fixed elements, (3) elements remain. Their subsets are \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). After excluding two fixed elements, (3) elements remain. Their subsets are \(2^3=8\).
Step 3
Exam Tip
दो निश्चित तत्व हटाने पर (3) तत्व बचते हैं। इनके उपसमुच्चय \(2^3=8\) होंगे।
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यदि \(U={x:x\in \mathbb{N}, x\leq 18}\) और \(A={x:x\) (4) का गुणज है(}), तो (A') में कितने तत्व होंगे?
If \(U={x:x\in \mathbb{N}, x\leq 18}\) and \(A={x:x\) is a multiple of (4)(}), how many elements are in (A')?
#sets
#universal set
#multiples
#class 11
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A (4)
B (12)
C (14)
D (18)
Explanation opens after your attempt
Step 1
Concept
\(A=\{4,8,12,16\}\), so (n(A)=4). Since (U) has (18) elements, (n(A')=18-4=14).
Step 2
Why this answer is correct
The correct answer is C. (14). \(A=\{4,8,12,16\}\), so (n(A)=4). Since (U) has (18) elements, (n(A')=18-4=14).
Step 3
Exam Tip
\(A=\{4,8,12,16\}\) है इसलिए (n(A)=4)। (U) में (18) तत्व हैं अतः (n(A')=18-4=14)।
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यदि \(A=\varnothing\) है, तो (\mathcal{P}(A)) के बारे में कौन सा कथन सही है?
If \(A=\varnothing\), which statement about (\mathcal{P}(A)) is correct?
#sets
#empty set
#power set
#class 11
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A (\mathcal{P}(A)=\varnothing)
B (\mathcal{P}(A)={\varnothing})
C (\mathcal{P}(A)={0})
D (\mathcal{P}(A)={A,{A}})
Explanation opens after your attempt
Correct Answer
B. (\mathcal{P}(A)={\varnothing})
Step 1
Concept
The only subset of the empty set is the empty set itself. Therefore (\mathcal{P}\(\varnothing\)={\varnothing}).
Step 2
Why this answer is correct
The correct answer is B. (\mathcal{P}(A)={\varnothing}). The only subset of the empty set is the empty set itself. Therefore (\mathcal{P}\(\varnothing\)={\varnothing}).
Step 3
Exam Tip
रिक्त समुच्चय का एकमात्र उपसमुच्चय वही रिक्त समुच्चय है। इसलिए (\mathcal{P}\(\varnothing\)={\varnothing})।
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यदि (A=U) है, तो (\mathcal{P}(A)) और (\mathcal{P}(U)) के बारे में कौन सा कथन सही है?
If (A=U), which statement about (\mathcal{P}(A)) and (\mathcal{P}(U)) is correct?
#sets
#power set
#equal sets
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A (\mathcal{P}(A)=\mathcal{P}(U))
B (\mathcal{P}(A)=\varnothing)
C (\mathcal{P}\((U)\subset A\))
D \(\mathcal{P}(A)\cap \mathcal{P}(U)=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. (\mathcal{P}(A)=\mathcal{P}(U))
Step 1
Concept
Equal sets have exactly the same subsets. Therefore their power sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (\mathcal{P}(A)=\mathcal{P}(U)). Equal sets have exactly the same subsets. Therefore their power sets are equal.
Step 3
Exam Tip
समान समुच्चयों के सभी उपसमुच्चय भी समान होते हैं। इसलिए उनके घात समुच्चय बराबर होंगे।
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यदि \(A=\varnothing\) और सार्वत्रिक समुच्चय \(U=\{2,4,6\}\) है, तो (A') क्या होगा?
If \(A=\varnothing\) and the universal set is \(U=\{2,4,6\}\), what is (A')?
#sets
#empty set
#complement
#class 11
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A \(\varnothing\)
B ({2,4,6})
C ({0})
D (\mathcal{P}(U))
Explanation opens after your attempt
Correct Answer
B. ({2,4,6})
Step 1
Concept
The empty set has no elements. Therefore its complement is the whole (U).
Step 2
Why this answer is correct
The correct answer is B. ({2,4,6}). The empty set has no elements. Therefore its complement is the whole (U).
Step 3
Exam Tip
रिक्त समुच्चय में कोई तत्व नहीं होता। इसलिए उसका पूरक पूरा (U) होगा।
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यदि \(A=\{3,6,9,12\}\) है, तो (\mathcal{P}(A)) में कितने अरिक्त तत्व होंगे?
If \(A=\{3,6,9,12\}\), how many non-empty elements are there in (\mathcal{P}(A))?
#sets
#power set
#non empty subsets
#class 11
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A (14)
B (15)
C (16)
D (4)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\). Only \(\varnothing\) is empty so there are (15) non-empty subsets.
Step 2
Why this answer is correct
The correct answer is B. (15). Total subsets are \(2^4=16\). Only \(\varnothing\) is empty so there are (15) non-empty subsets.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं। केवल \(\varnothing\) रिक्त है इसलिए अरिक्त उपसमुच्चय (15) होंगे।
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यदि \(A=\{a,e,i,o,u\}\) है, तो (\mathcal{P}(A)) में कितने एक-तत्वीय समुच्चय होंगे?
If \(A=\{a,e,i,o,u\}\), how many singleton sets are in (\mathcal{P}(A))?
#sets
#power set
#singleton subsets
#class 11
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A (4)
B (5)
C (10)
D (32)
Explanation opens after your attempt
Step 1
Concept
Each original element forms one singleton subset. From (5) elements, there are (5) singleton subsets.
Step 2
Why this answer is correct
The correct answer is B. (5). Each original element forms one singleton subset. From (5) elements, there are (5) singleton subsets.
Step 3
Exam Tip
हर मूल तत्व से एक एक-तत्वीय उपसमुच्चय बनता है। (5) तत्वों से (5) singleton subsets बनेंगे।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में ठीक (4) तत्व वाले समुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5,6\}\), how many sets in (\mathcal{P}(A)) have exactly (4) elements?
#sets
#power set
#combination
#class 11
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A (12)
B (15)
C (20)
D (24)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly (4) elements is \(\binom{6}{4}=15\). Every such subset is counted in the power set.
Step 2
Why this answer is correct
The correct answer is B. (15). The number of ways to choose exactly (4) elements is \(\binom{6}{4}=15\). Every such subset is counted in the power set.
Step 3
Exam Tip
ठीक (4) तत्व चुनने की संख्या \(\binom{6}{4}=15\) है। ऐसे हर उपसमुच्चय को घात समुच्चय में गिना जाता है।
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यदि \(A=\{2,3\}\) है, तो (\mathcal{P}(A)) का सही रूप कौन सा है?
If \(A=\{2,3\}\), which is the correct form of (\mathcal{P}(A))?
#sets
#power set
#representation
#class 11
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A \({\varnothing,{2},{3},{2,3}}\)
B ({2,3,{2,3}})
C ({{2},{3}})
D \({\varnothing,2,3}\)
Explanation opens after your attempt
Correct Answer
A. \({\varnothing,{2},{3},{2,3}}\)
Step 1
Concept
A two-element set has four subsets. In the power set, (2) and (3) appear as ({2}) and ({3}), not alone.
Step 2
Why this answer is correct
The correct answer is A. \({\varnothing,{2},{3},{2,3}}\). A two-element set has four subsets. In the power set, (2) and (3) appear as ({2}) and ({3}), not alone.
Step 3
Exam Tip
दो तत्वों वाले समुच्चय के चार उपसमुच्चय होते हैं। घात समुच्चय में (2) और (3) अकेले नहीं बल्कि ({2}) और ({3}) आते हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6\}\) और \(B=\{4,6,8\}\) हैं, तो \(A'\cap B'\) क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6\}\), and \(B=\{4,6,8\}\), what is \(A'\cap B'\)?
#sets
#complement
#intersection
#class 11
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A ({1,3,5,7})
B ({4,6})
C ({1,2,3,5,7})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,7})
Step 1
Concept
(A'={1,3,5,7,8}) and (B'={1,2,3,5,7}). Their intersection is ({1,3,5,7}).
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,7}). (A'={1,3,5,7,8}) and (B'={1,2,3,5,7}). Their intersection is ({1,3,5,7}).
Step 3
Exam Tip
(A'={1,3,5,7,8}) और (B'={1,2,3,5,7}) हैं। इनका प्रतिच्छेद ({1,3,5,7}) है।
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यदि \(U=\{p,q,r,s,t\}\), \(A=\{p,q,r\}\) और \(B=\{r,s\}\) हैं, तो (\(A\cap B\)') क्या होगा?
If \(U=\{p,q,r,s,t\}\), \(A=\{p,q,r\}\), and \(B=\{r,s\}\), what is (\(A\cap B\)')?
#sets
#universal set
#complement of intersection
#class 11
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A ({r})
B ({p,q,s,t})
C ({p,q,r,s})
D ({t})
Explanation opens after your attempt
Correct Answer
B. ({p,q,s,t})
Step 1
Concept
\(A\cap B={r}\). Removing (r) from (U) gives ({p,q,s,t}).
Step 2
Why this answer is correct
The correct answer is B. ({p,q,s,t}). \(A\cap B={r}\). Removing (r) from (U) gives ({p,q,s,t}).
Step 3
Exam Tip
\(A\cap B={r}\) है। (U) से (r) हटाने पर ({p,q,s,t}) मिलता है।
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यदि (n(A)=6) और (n(B)=4) है, तो (n(\mathcal{P}(A)) : n(\mathcal{P}(B))) क्या होगा?
If (n(A)=6) and (n(B)=4), what is (n(\mathcal{P}(A)) : n(\mathcal{P}(B)))?
#sets
#power set
#ratio
#class 11
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A (3:2)
B (4:1)
C (6:4)
D (16:1)
Explanation opens after your attempt
Step 1
Concept
(n(\mathcal{P}(A))=64) and (n(\mathcal{P}(B))=16). The ratio is (64:16=4:1).
Step 2
Why this answer is correct
The correct answer is B. (4:1). (n(\mathcal{P}(A))=64) and (n(\mathcal{P}(B))=16). The ratio is (64:16=4:1).
Step 3
Exam Tip
(n(\mathcal{P}(A))=64) और (n(\mathcal{P}(B))=16) हैं। अनुपात (64:16=4:1) होगा।
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यदि \(A={x:x\in \mathbb{N}, 5<x<11}\) है, तो (n(\mathcal{P}(A))) क्या होगा?
If \(A={x:x\in \mathbb{N}, 5<x<11}\), what is (n(\mathcal{P}(A)))?
#sets
#power set
#set builder
#class 11
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A (16)
B (32)
C (64)
D (128)
Explanation opens after your attempt
Step 1
Concept
\(A=\{6,7,8,9,10\}\), so (n(A)=5). Hence (n(\mathcal{P}(A))=25 =32).
Step 2
Why this answer is correct
The correct answer is B. (32). \(A=\{6,7,8,9,10\}\), so (n(A)=5). Hence (n(\mathcal{P}(A))=25 =32).
Step 3
Exam Tip
\(A=\{6,7,8,9,10\}\) है इसलिए (n(A)=5)। अतः (n(\mathcal{P}(A))=25 =32)।
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यदि \(A={x:x\) शब्द (MOON) के अलग-अलग अक्षर हैं(}), तो (\mathcal{P}(A)) में कितने तत्व होंगे?
If \(A={x:x\) is a distinct letter of the word (MOON)(}), how many elements will (\mathcal{P}(A)) have?
#sets
#power set
#distinct elements
#class 11
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
The distinct letters of (MOON) are (M,O,N), so (n(A)=3). Therefore the power set has \(2^3=8\) elements.
Step 2
Why this answer is correct
The correct answer is B. (8). The distinct letters of (MOON) are (M,O,N), so (n(A)=3). Therefore the power set has \(2^3=8\) elements.
Step 3
Exam Tip
(MOON) के अलग-अलग अक्षर (M,O,N) हैं इसलिए (n(A)=3)। अतः घात समुच्चय में \(2^3=8\) तत्व होंगे।
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यदि \(A=\{4,4,5,6,6,7\}\) को समुच्चय माना जाए, तो (n(\mathcal{P}(A))) कितना होगा?
If \(A=\{4,4,5,6,6,7\}\) is considered as a set, what is (n(\mathcal{P}(A)))?
#sets
#power set
#repeated elements
#class 11
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
Repeated elements are counted once in a set, so \(A=\{4,5,6,7\}\). Hence (n(\mathcal{P}(A))=24 =16).
Step 2
Why this answer is correct
The correct answer is B. (16). Repeated elements are counted once in a set, so \(A=\{4,5,6,7\}\). Hence (n(\mathcal{P}(A))=24 =16).
Step 3
Exam Tip
समुच्चय में दोहराए गए तत्व एक बार गिने जाते हैं इसलिए \(A=\{4,5,6,7\}\)। इसलिए (n(\mathcal{P}(A))=24 =16)।
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यदि \(A=\{a,b,c,d,e,f,g\}\) है, तो (\mathcal{P}(A)) में ठीक (6) तत्व वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d,e,f,g\}\), how many subsets with exactly (6) elements are in (\mathcal{P}(A))?
#sets
#power set
#combination
#class 11
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A (6)
B (7)
C (14)
D (21)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose (6) elements from (7) is \(\binom{7}{6}=7\). Each choice is a subset.
Step 2
Why this answer is correct
The correct answer is B. (7). The number of ways to choose (6) elements from (7) is \(\binom{7}{6}=7\). Each choice is a subset.
Step 3
Exam Tip
(7) में से (6) तत्व चुनने के तरीके \(\binom{7}{6}=7\) हैं। हर चयन एक उपसमुच्चय है।
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यदि किसी समुच्चय (A) के कुल उपसमुच्चय (2048) हैं, तो (A) में कितने तत्व होंगे?
If a set (A) has (2048) total subsets, how many elements does (A) have?
#sets
#power set
#total subsets
#class 11
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^n\), and \(2048=2^{11}\). Therefore (A) has (11) elements.
Step 2
Why this answer is correct
The correct answer is C. (11). Total subsets are \(2^n\), and \(2048=2^{11}\). Therefore (A) has (11) elements.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^n\) होते हैं और \(2048=2^{11}\)। इसलिए (A) में (11) तत्व होंगे।
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यदि \(A\subseteq B\subseteq U\) है, तो पूरकों के लिए कौन सा संबंध सही है?
If \(A\subseteq B\subseteq U\), which relation is correct for complements?
#sets
#complement
#subset relation
#class 11
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A \(A'\subseteq B'\)
B \(B'\subseteq A'\)
C (A'=B')
D \(A'\cap B'=\varnothing\)
Explanation opens after your attempt
Correct Answer
B. \(B'\subseteq A'\)
Step 1
Concept
The complement of the smaller set is larger. If \(A\subseteq B\), then \(B'\subseteq A'\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(B'\subseteq A'\). The complement of the smaller set is larger. If \(A\subseteq B\), then \(B'\subseteq A'\) is correct.
Step 3
Exam Tip
छोटे समुच्चय का पूरक बड़ा होता है। \(A\subseteq B\) होने पर \(B'\subseteq A'\) सही है।
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यदि \(U=\{2,4,6,8,10,12\}\) और (A'={4,10}) है, तो (A) क्या होगा?
If \(U=\{2,4,6,8,10,12\}\) and (A'={4,10}), what is (A)?
#sets
#universal set
#find set from complement
#class 11
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A ({2,6,8,12})
B ({4,10})
C ({2,4,6,8})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({2,6,8,12})
Step 1
Concept
Elements of (A') are not in (A). Removing (4,10) from (U) gives \(A=\{2,6,8,12\}\).
Step 2
Why this answer is correct
The correct answer is A. ({2,6,8,12}). Elements of (A') are not in (A). Removing (4,10) from (U) gives \(A=\{2,6,8,12\}\).
Step 3
Exam Tip
(A') के तत्व (A) में नहीं होते। (U) से (4,10) हटाने पर \(A=\{2,6,8,12\}\) मिलता है।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जिनमें (2) हो लेकिन (5) न हो?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) contain (2) but not (5)?
#sets
#power set
#conditional subsets
#class 11
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
Including (2) and excluding (5) are fixed. For the remaining (1,3,4), there are \(2^3=8\) choices.
Step 2
Why this answer is correct
The correct answer is B. (8). Including (2) and excluding (5) are fixed. For the remaining (1,3,4), there are \(2^3=8\) choices.
Step 3
Exam Tip
(2) रखना और (5) हटाना निश्चित है। बचे (1,3,4) के लिए \(2^3=8\) विकल्प हैं।
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यदि \(A=\{p,q,r,s,t,u\}\) है, तो (A) के ऐसे कितने उपसमुच्चय हैं जिनमें कम से कम (5) तत्व हों?
If \(A=\{p,q,r,s,t,u\}\), how many subsets of (A) contain at least (5) elements?
#sets
#power set
#at least subsets
#class 11
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A (6)
B (7)
C (12)
D (22)
Explanation opens after your attempt
Step 1
Concept
At least (5) means (5) or (6) elements. The number is \(\binom{6}{5}+\binom{6}{6}=6+1=7\).
Step 2
Why this answer is correct
The correct answer is B. (7). At least (5) means (5) or (6) elements. The number is \(\binom{6}{5}+\binom{6}{6}=6+1=7\).
Step 3
Exam Tip
कम से कम (5) का अर्थ (5) या (6) तत्व है। संख्या \(\binom{6}{5}+\binom{6}{6}=6+1=7\) है।
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यदि \(A=\{1,2,3,4,5,6,7\}\) है, तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (3) और (7) दोनों हों?
If \(A=\{1,2,3,4,5,6,7\}\), how many subsets of (A) contain both (3) and (7)?
#sets
#power set
#subset counting
#class 11
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A (16)
B (32)
C (64)
D (128)
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Step 1
Concept
Including (3) and (7) is fixed and the remaining (5) elements are free. So the number is \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. (32). Including (3) and (7) is fixed and the remaining (5) elements are free. So the number is \(2^5=32\).
Step 3
Exam Tip
(3) और (7) को रखना तय है और बाकी (5) तत्व स्वतंत्र हैं। इसलिए संख्या \(2^5=32\) होगी।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (1), (2) और (3) में से कोई भी न हो?
If \(A=\{1,2,3,4,5,6\}\), how many subsets of (A) contain none of (1), (2), and (3)?
#sets
#power set
#restricted subsets
#class 11
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
After excluding (1,2,3), the elements (4,5,6) remain. Their subsets are \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). After excluding (1,2,3), the elements (4,5,6) remain. Their subsets are \(2^3=8\).
Step 3
Exam Tip
(1,2,3) हटाने पर (4,5,6) बचते हैं। इनके \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A=\{g,h,i,j\}\) है, तो (\mathcal{P}(A)) के कितने तत्व (A) के उचित उपसमुच्चय भी हैं?
If \(A=\{g,h,i,j\}\), how many elements of (\mathcal{P}(A)) are also proper subsets of (A)?
#sets
#power set
#proper subset
#class 11
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A (14)
B (15)
C (16)
D (4)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^4=16\) elements. For proper subsets, remove only (A) itself, so (15) remain.
Step 2
Why this answer is correct
The correct answer is B. (15). (\mathcal{P}(A)) has \(2^4=16\) elements. For proper subsets, remove only (A) itself, so (15) remain.
Step 3
Exam Tip
(\mathcal{P}(A)) में कुल \(2^4=16\) तत्व हैं। उचित उपसमुच्चय के लिए केवल (A) को हटाते हैं, इसलिए (15) बचते हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{1,3,5,7\}\) और \(B=\{2,3,6,7\}\) हैं, तो (\(A\cup B\)') क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{1,3,5,7\}\), and \(B=\{2,3,6,7\}\), what is (\(A\cup B\)')?
#sets
#universal set
#union complement
#class 11
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A ({4,8,9,10})
B ({1,2,3,5,6,7})
C ({3,7})
D ({4,8,10})
Explanation opens after your attempt
Correct Answer
A. ({4,8,9,10})
Step 1
Concept
\(A\cup B={1,2,3,5,6,7}\). Removing it from (U) gives ({4,8,9,10}).
Step 2
Why this answer is correct
The correct answer is A. ({4,8,9,10}). \(A\cup B={1,2,3,5,6,7}\). Removing it from (U) gives ({4,8,9,10}).
Step 3
Exam Tip
\(A\cup B={1,2,3,5,6,7}\) है। (U) से इसे हटाने पर ({4,8,9,10}) मिलता है।
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यदि \(A=\{2,4,6\}\) और \(B=\{6,8,10\}\) हैं, तो (\mathcal{P}\(A\cup B\)) में कितने तत्व होंगे?
If \(A=\{2,4,6\}\) and \(B=\{6,8,10\}\), how many elements will (\mathcal{P}\(A\cup B\)) have?
#sets
#power set
#union
#class 11
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A (16)
B (32)
C (64)
D (128)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={2,4,6,8,10}\), so it has (5) elements. Hence its power set has \(2^5=32\) elements.
Step 2
Why this answer is correct
The correct answer is B. (32). \(A\cup B={2,4,6,8,10}\), so it has (5) elements. Hence its power set has \(2^5=32\) elements.
Step 3
Exam Tip
\(A\cup B={2,4,6,8,10}\) है इसलिए इसमें (5) तत्व हैं। अतः घात समुच्चय में \(2^5=32\) तत्व होंगे।
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यदि \(U={x:x\in \mathbb{N}, x\leq 30}\) और \(A={x:x\) (30) का धनात्मक भाजक है(}) है, तो (n(A')) कितना होगा?
If \(U={x:x\in \mathbb{N}, x\leq 30}\) and \(A={x:x\) is a positive divisor of (30)(}), what is (n(A'))?
#sets
#universal set
#divisors complement
#class 11
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A (8)
B (20)
C (22)
D (30)
Explanation opens after your attempt
Step 1
Concept
The divisors of (30) are (1,2,3,5,6,10,15,30), so (n(A)=8). Hence (n(A')=30-8=22).
Step 2
Why this answer is correct
The correct answer is C. (22). The divisors of (30) are (1,2,3,5,6,10,15,30), so (n(A)=8). Hence (n(A')=30-8=22).
Step 3
Exam Tip
(30) के भाजक (1,2,3,5,6,10,15,30) हैं इसलिए (n(A)=8)। अतः (n(A')=30-8=22)।
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यदि \(A=\{1,2,3\}\) है, तो (\mathcal{P}(A)) के कितने तत्वों में कम से कम एक तत्व है?
If \(A=\{1,2,3\}\), how many elements of (\mathcal{P}(A)) have at least one element?
#sets
#power set
#non empty subsets
#class 11
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A (6)
B (7)
C (8)
D (3)
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Step 1
Concept
Subsets with at least one element are non-empty subsets. From total \(2^3=8\), removing \(\varnothing\) leaves (7).
Step 2
Why this answer is correct
The correct answer is B. (7). Subsets with at least one element are non-empty subsets. From total \(2^3=8\), removing \(\varnothing\) leaves (7).
Step 3
Exam Tip
कम से कम एक तत्व वाले उपसमुच्चय अरिक्त उपसमुच्चय होते हैं। कुल \(2^3=8\) में से \(\varnothing\) हटाने पर (7) बचते हैं।
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