यदि (n(\mathcal{P}(A))=64) और (U) में (10) तत्व हैं, तो (A') में कितने तत्व होंगे?
If (n(\mathcal{P}(A))=64) and (U) has (10) elements, how many elements are in (A')?
#sets
#power-set
#complement
#cardinality
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A (4)
B (6)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
Since \(2^{n(A)}=64\), (n(A)=6), so (n(A')=10-6=4). In exams, identify the universal set first.
Step 2
Why this answer is correct
The correct answer is A. (4). Since \(2^{n(A)}=64\), (n(A)=6), so (n(A')=10-6=4). In exams, identify the universal set first.
Step 3
Exam Tip
क्योंकि \(2^{n(A)}=64\), इसलिए (n(A)=6) और (n(A')=10-6=4)। परीक्षा में पहले आधार समुच्चय को पहचानें।
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यदि \(U={1,2,\ldots,16}\), \(A={x:x\) (16) का भाजक है(}) और (B=A'), तो (n(\mathcal{P}(B))) कितना है?
If \(U={1,2,\ldots,16}\), \(A={x:x\) is a divisor of (16)(}), and (B=A'), what is (n(\mathcal{P}(B)))?
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#power-set
#universal-set
#complement
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A (2048)
B (1024)
C (4096)
D (512)
Explanation opens after your attempt
Step 1
Concept
The divisors of (16) are (1,2,4,8,16), so (A') has (11) elements. Hence (n(\mathcal{P}(B))=2^{11}=2048).
Step 2
Why this answer is correct
The correct answer is A. (2048). The divisors of (16) are (1,2,4,8,16), so (A') has (11) elements. Hence (n(\mathcal{P}(B))=2^{11}=2048).
Step 3
Exam Tip
(16) के भाजक (1,2,4,8,16) हैं, इसलिए (A') में (11) तत्व हैं। अतः (n(\mathcal{P}(B))=2^{11}=2048)।
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यदि \(A=\{1,{2},3\}\) है, तो कौन सा (\mathcal{P}(A)) का तत्व है?
If \(A=\{1,{2},3\}\), which one is an element of (\mathcal{P}(A))?
#sets
#power-set
#subset
#concept
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A ({1,{2}})
B (2)
C ({{3}})
D ({2,3})
Explanation opens after your attempt
Correct Answer
A. ({1,{2}})
Step 1
Concept
Every element of a power set is a subset of the original set. ({1,{2}}) contains only elements of (A).
Step 2
Why this answer is correct
The correct answer is A. ({1,{2}}). Every element of a power set is a subset of the original set. ({1,{2}}) contains only elements of (A).
Step 3
Exam Tip
घात समुच्चय का प्रत्येक तत्व मूल समुच्चय का उपसमुच्चय होता है। ({1,{2}}) में केवल (A) के तत्व हैं।
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यदि \(A={\varnothing,{1},2}\), तो कौन सा (\mathcal{P}(A)) का तत्व नहीं है?
If \(A={\varnothing,{1},2}\), which one is not an element of (\mathcal{P}(A))?
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#power-set
#empty-set
#membership
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A \({\varnothing,2}\)
B ({{1},2})
C ({1,2})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
C. ({1,2})
Step 1
Concept
({1,2}) contains (1), but (1) itself is not an element of (A). Elements of a power set are always subsets of the original set.
Step 2
Why this answer is correct
The correct answer is C. ({1,2}). ({1,2}) contains (1), but (1) itself is not an element of (A). Elements of a power set are always subsets of the original set.
Step 3
Exam Tip
({1,2}) में (1) है, लेकिन (1) स्वयं (A) का तत्व नहीं है। घात समुच्चय के तत्व हमेशा मूल समुच्चय के उपसमुच्चय होते हैं।
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यदि \(U={1,2,\ldots,12}\) और \(A={x:x\) (12) का भाजक है(}), तो (A') क्या है?
If \(U={1,2,\ldots,12}\) and \(A={x:x\) is a divisor of (12)(}), what is (A')?
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#universal-set
#complement
#divisors
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A ({5,7,8,9,10,11})
B ({1,2,3,4,6,12})
C ({2,3,4,6})
D ({5,7,9,11})
Explanation opens after your attempt
Correct Answer
A. ({5,7,8,9,10,11})
Step 1
Concept
The divisors of (12) are (1,2,3,4,6,12), so the remaining elements form the complement. Always take complement with respect to (U).
Step 2
Why this answer is correct
The correct answer is A. ({5,7,8,9,10,11}). The divisors of (12) are (1,2,3,4,6,12), so the remaining elements form the complement. Always take complement with respect to (U).
Step 3
Exam Tip
(12) के भाजक (1,2,3,4,6,12) हैं, इसलिए बाकी तत्व पूरक में आएंगे। परीक्षा में पूरक हमेशा (U) के संदर्भ में लें।
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यदि (n(\mathcal{P}(A))=32) और (n(U)=13), तो (A') के (2)-तत्वीय उपसमुच्चयों की संख्या कितनी है?
If (n(\mathcal{P}(A))=32) and (n(U)=13), how many (2)-element subsets does (A') have?
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#complement
#combination
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A (21)
B (28)
C (36)
D (15)
Explanation opens after your attempt
Step 1
Concept
Since \(2^{n(A)}=32\), (n(A)=5) and (n(A')=8). The number of (2)-element subsets of (A') is \(\binom{8}{2}=28\).
Step 2
Why this answer is correct
The correct answer is B. (28). Since \(2^{n(A)}=32\), (n(A)=5) and (n(A')=8). The number of (2)-element subsets of (A') is \(\binom{8}{2}=28\).
Step 3
Exam Tip
\(2^{n(A)}=32\), इसलिए (n(A)=5) और (n(A')=8)। (A') के (2)-तत्वीय उपसमुच्चय \(\binom{8}{2}=28\) होंगे।
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यदि (\mathcal{P}(A)=\mathcal{P}(B)), तो कौन सा निष्कर्ष हमेशा सही है?
If (\mathcal{P}(A)=\mathcal{P}(B)), which conclusion is always true?
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#reasoning
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A (A=B)
B \(A\subset B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
If two sets have exactly the same subsets, the original sets must be equal. This is an important power set result for exams.
Step 2
Why this answer is correct
The correct answer is A. (A=B). If two sets have exactly the same subsets, the original sets must be equal. This is an important power set result for exams.
Step 3
Exam Tip
यदि दो समुच्चयों के सभी उपसमुच्चय समान हैं, तो मूल समुच्चय भी समान होंगे। यह घात समुच्चय की एक महत्वपूर्ण परीक्षा-युक्ति है।
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यदि \(U={1,2,\ldots,20}\), (A) (4) के गुणजों का समुच्चय और (B) (6) के गुणजों का समुच्चय है, तो (n(\(A\cup B\)')) कितना है?
If \(U={1,2,\ldots,20}\), (A) is the set of multiples of (4), and (B) is the set of multiples of (6), what is (n(\(A\cup B\)'))?
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#universal-set
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#inclusion-exclusion
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A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
There are (5) multiples of (4) and (3) multiples of (6), with (1) common multiple of (12). So the union has (7) elements and the complement has (20-7=13) elements.
Step 2
Why this answer is correct
The correct answer is D. (13). There are (5) multiples of (4) and (3) multiples of (6), with (1) common multiple of (12). So the union has (7) elements and the complement has (20-7=13) elements.
Step 3
Exam Tip
(4) के (5) और (6) के (3) गुणज हैं, जबकि (12) का केवल (1) साझा गुणज है। इसलिए संघ में (7) तत्व और पूरक में (20-7=13) तत्व होंगे।
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किसी समुच्चय (A) के लिए (n(\mathcal{P}(\mathcal{P}(A)))=16) है, तो (n(A)) कितना है?
For a set (A), (n(\mathcal{P}(\mathcal{P}(A)))=16). What is (n(A))?
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#cardinality
#double-power-set
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
Here \(2^{2^{n(A)}}=16=2^4\), so \(2^{n(A)}=4\) and (n(A)=2). Apply the power set formula twice.
Step 2
Why this answer is correct
The correct answer is B. (2). Here \(2^{2^{n(A)}}=16=2^4\), so \(2^{n(A)}=4\) and (n(A)=2). Apply the power set formula twice.
Step 3
Exam Tip
यहां \(2^{2^{n(A)}}=16=2^4\), इसलिए \(2^{n(A)}=4\) और (n(A)=2)। घात समुच्चय पर घात समुच्चय में दो बार सूत्र लगाएं।
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यदि \(A=\{a,b,c,d,e\}\), तो \(\mathcal{P}(A)\) में ऐसे कितने उपसमुच्चय हैं जिनमें (a) है और (b) तथा (c) दोनों नहीं हैं?
If \(A=\{a,b,c,d,e\}\), how many subsets in \(\mathcal{P}(A)\) contain (a) and contain neither (b) nor (c)?
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#power-set
#counting
#conditional-subsets
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
(a) is fixed and (b,c) are excluded, so only (d,e) are free. Therefore \(2^2=4\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (4). (a) is fixed and (b,c) are excluded, so only (d,e) are free. Therefore \(2^2=4\) subsets are possible.
Step 3
Exam Tip
(a) निश्चित है और (b,c) बाहर हैं, इसलिए केवल (d,e) स्वतंत्र हैं। कुल \(2^2=4\) उपसमुच्चय बनेंगे।
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यदि \(U=\{a,b,c,d,e\}\), \(A=\{a,c,e\}\) और \(B=\{b,e\}\), तो (\(A\cup B\)') क्या है?
If \(U=\{a,b,c,d,e\}\), \(A=\{a,c,e\}\), and \(B=\{b,e\}\), what is (\(A\cup B\)')?
#sets
#universal-set
#union
#complement
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A ({d})
B ({a,b,c,e})
C ({b,d})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={a,b,c,e}\), so the only remaining element in (U) is (d). Find the union before taking complement.
Step 2
Why this answer is correct
The correct answer is A. ({d}). \(A\cup B={a,b,c,e}\), so the only remaining element in (U) is (d). Find the union before taking complement.
Step 3
Exam Tip
\(A\cup B={a,b,c,e}\), इसलिए (U) में बचा तत्व (d) है। पूरक निकालने से पहले संघ पूरा करें।
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यदि (A) के ठीक (5) उचित उपसमुच्चय हैं, तो (\mathcal{P}(A)) में कितने तत्व होंगे?
If (A) has exactly (5) proper subsets, how many elements are in (\mathcal{P}(A))?
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#proper-subset
#trick
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A (6)
B (5)
C (8)
D (4)
Explanation opens after your attempt
Step 1
Concept
The number of proper subsets is \(2^{n(A)}-1\), so \(2^{n(A)}=6\). The power set also includes the original set itself.
Step 2
Why this answer is correct
The correct answer is A. (6). The number of proper subsets is \(2^{n(A)}-1\), so \(2^{n(A)}=6\). The power set also includes the original set itself.
Step 3
Exam Tip
उचित उपसमुच्चय की संख्या \(2^{n(A)}-1\) होती है, इसलिए \(2^{n(A)}=6\)। घात समुच्चय में मूल समुच्चय भी शामिल होता है।
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पिछले प्रश्न जैसी स्थिति में यदि किसी वास्तविक समुच्चय के उचित उपसमुच्चय (5) बताए जाएं, तो सही निष्कर्ष क्या है?
In a situation where a real finite set is said to have (5) proper subsets, what is the correct conclusion?
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#validity
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A ऐसा कोई सीमित समुच्चय नहीं है / No such finite set exists
B समुच्चय में (2) तत्व हैं / The set has (2) elements
C समुच्चय में (3) तत्व हैं / The set has (3) elements
D समुच्चय रिक्त है / The set is empty
Explanation opens after your attempt
Correct Answer
A. ऐसा कोई सीमित समुच्चय नहीं है / No such finite set exists
Step 1
Concept
Because \(2^n-1=5\) gives \(2^n=6\), which is not a power of (2). In such questions, also check whether the number is possible.
Step 2
Why this answer is correct
The correct answer is A. ऐसा कोई सीमित समुच्चय नहीं है / No such finite set exists. Because \(2^n-1=5\) gives \(2^n=6\), which is not a power of (2). In such questions, also check whether the number is possible.
Step 3
Exam Tip
क्योंकि \(2^n-1=5\) से \(2^n=6\) मिलता है, जो (2) की घात नहीं है। ऐसे प्रश्न में संख्या की सम्भवता भी जांचें।
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यदि \(U={1,2,\ldots,20}\) और (A) सभी अभाज्य संख्याओं का समुच्चय है, तो (n(\mathcal{P}(A'))) कितना है?
If \(U={1,2,\ldots,20}\) and (A) is the set of all prime numbers, what is (n(\mathcal{P}(A')))?
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#power-set
#universal-set
#prime-numbers
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A (4096)
B (256)
C (1024)
D (8192)
Explanation opens after your attempt
Step 1
Concept
There are (8) primes from (1) to (20), so (A') has (12) elements. Hence (n(\mathcal{P}(A'))=2^{12}=4096).
Step 2
Why this answer is correct
The correct answer is A. (4096). There are (8) primes from (1) to (20), so (A') has (12) elements. Hence (n(\mathcal{P}(A'))=2^{12}=4096).
Step 3
Exam Tip
(1) से (20) तक (8) अभाज्य संख्याएं हैं, इसलिए (A') में (12) तत्व हैं। अतः (n(\mathcal{P}(A'))=2^{12}=4096)।
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यदि \(A=\{1,2\}\), तो निम्न में से कौन सा कथन गलत है?
If \(A=\{1,2\}\), which statement is false?
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A \(1\in\mathcal{P}(A)\)
B \({1}\in\mathcal{P}(A)\)
C \(\varnothing\in\mathcal{P}(A)\)
D \(A\in\mathcal{P}(A)\)
Explanation opens after your attempt
Correct Answer
A. \(1\in\mathcal{P}(A)\)
Step 1
Concept
Elements of (\mathcal{P}(A)) are subsets, not ordinary elements. Therefore \(1\in\mathcal{P}(A)\) is false.
Step 2
Why this answer is correct
The correct answer is A. \(1\in\mathcal{P}(A)\). Elements of (\mathcal{P}(A)) are subsets, not ordinary elements. Therefore \(1\in\mathcal{P}(A)\) is false.
Step 3
Exam Tip
(\mathcal{P}(A)) के तत्व उपसमुच्चय होते हैं, सामान्य तत्व नहीं। इसलिए \(1\in\mathcal{P}(A)\) गलत है।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,3,5,7\}\) और \(B=\{2,3,5,8\}\), तो \(A'\cap B'\) क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,3,5,7\}\), and \(B=\{2,3,5,8\}\), what is \(A'\cap B'\)?
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#universal-set
#complement
#intersection
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A ({4,6})
B ({3,5})
C ({1,2,7,8})
D ({1,2,3,5,7,8})
Explanation opens after your attempt
Correct Answer
A. ({4,6})
Step 1
Concept
(A'={2,4,6,8}) and (B'={1,4,6,7}), so the intersection is ({4,6}). Find both complements separately.
Step 2
Why this answer is correct
The correct answer is A. ({4,6}). (A'={2,4,6,8}) and (B'={1,4,6,7}), so the intersection is ({4,6}). Find both complements separately.
Step 3
Exam Tip
(A'={2,4,6,8}) और (B'={1,4,6,7}), अतः प्रतिच्छेद ({4,6}) है। ऐसे प्रश्न में दोनों पूरक अलग निकालें।
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यदि (n(A)=m) और (n(\mathcal{P}(A))=n(A)+12), तो (m) क्या है?
If (n(A)=m) and (n(\mathcal{P}(A))=n(A)+12), what is (m)?
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#equation
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A (4)
B (3)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The equation is \(2^m=m+12\), and (m=4) gives (16=16). Testing small integers is a quick method.
Step 2
Why this answer is correct
The correct answer is A. (4). The equation is \(2^m=m+12\), and (m=4) gives (16=16). Testing small integers is a quick method.
Step 3
Exam Tip
समीकरण \(2^m=m+12\) है और (m=4) रखने पर (16=16) मिलता है। छोटी पूर्ण संख्याएं जांचना तेज तरीका है।
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यदि \(U=\mathbb{N}\) और \(A={x:x\) सम संख्या है(}), तो (A') क्या दर्शाता है?
If \(U=\mathbb{N}\) and \(A={x:x\) is an even number(}), what does (A') represent?
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#universal-set
#complement
#natural-numbers
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A सभी विषम प्राकृतिक संख्याएं / All odd natural numbers
B सभी सम पूर्णांक / All even integers
C सभी वास्तविक संख्याएं / All real numbers
D सभी ऋणात्मक संख्याएं / All negative numbers
Explanation opens after your attempt
Correct Answer
A. सभी विषम प्राकृतिक संख्याएं / All odd natural numbers
Step 1
Concept
When \(U=\mathbb{N}\), complement is taken only within natural numbers. Thus the complement of even natural numbers is odd natural numbers.
Step 2
Why this answer is correct
The correct answer is A. सभी विषम प्राकृतिक संख्याएं / All odd natural numbers. When \(U=\mathbb{N}\), complement is taken only within natural numbers. Thus the complement of even natural numbers is odd natural numbers.
Step 3
Exam Tip
जब \(U=\mathbb{N}\) है, तो पूरक केवल प्राकृतिक संख्याओं के अंदर लिया जाएगा। इसलिए सम प्राकृतिक संख्याओं का पूरक विषम प्राकृतिक संख्याएं हैं।
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यदि (A) में (3) तत्व हैं, तो (\mathcal{P}(A)) के कितने तत्व स्वयं (A) के एक-तत्वीय उपसमुच्चय हैं?
If (A) has (3) elements, how many elements of (\mathcal{P}(A)) are singleton subsets of (A)?
#sets
#power-set
#singleton
#subsets
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A (3)
B (1)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
A set with (3) elements has (3) singleton subsets. All of them are elements of (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is A. (3). A set with (3) elements has (3) singleton subsets. All of them are elements of (\mathcal{P}(A)).
Step 3
Exam Tip
तीन तत्वों वाले समुच्चय के एक-तत्वीय उपसमुच्चय (3) होते हैं। ये सभी (\mathcal{P}(A)) के तत्व हैं।
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यदि \(A=\{a,b,c,d\}\), तो (\mathcal{P}(A)) में ठीक (2) तत्वों वाले कितने उपसमुच्चय होंगे?
If \(A=\{a,b,c,d\}\), how many subsets with exactly (2) elements are in (\mathcal{P}(A))?
#sets
#power-set
#combination
#subsets
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A (6)
B (8)
C (4)
D (12)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly (2) elements is \(\binom{4}{2}=6\). The power set contains subsets of every size.
Step 2
Why this answer is correct
The correct answer is A. (6). The number of ways to choose exactly (2) elements is \(\binom{4}{2}=6\). The power set contains subsets of every size.
Step 3
Exam Tip
ठीक (2) तत्व चुनने के तरीके \(\binom{4}{2}=6\) हैं। घात समुच्चय में सभी आकारों के उपसमुच्चय होते हैं।
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यदि \(U={x:x\) (10) से छोटी धनात्मक पूर्ण संख्या है(}) और \(A={x:x^2<20}\), तो (A') क्या है?
If \(U={x:x\) is a positive integer less than (10)(}) and \(A={x:x^2<20}\), what is (A')?
#sets
#universal-set
#complement
#set-builder
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A ({5,6,7,8,9})
B ({1,2,3,4})
C ({4,5,6,7,8,9})
D ({6,7,8,9})
Explanation opens after your attempt
Correct Answer
A. ({5,6,7,8,9})
Step 1
Concept
\(U={1,\ldots,9}\), and \(x^2<20\) gives \(A=\{1,2,3,4\}\). The remaining set ({5,6,7,8,9}) is the complement.
Step 2
Why this answer is correct
The correct answer is A. ({5,6,7,8,9}). \(U={1,\ldots,9}\), and \(x^2<20\) gives \(A=\{1,2,3,4\}\). The remaining set ({5,6,7,8,9}) is the complement.
Step 3
Exam Tip
\(U={1,\ldots,9}\) और \(x^2<20\) के लिए \(A=\{1,2,3,4\}\) है। बाकी ({5,6,7,8,9}) पूरक है।
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यदि \(A=\{1,2,3\}\), तो (\mathcal{P}(A)) में कितने तत्व ऐसे हैं जिनमें (1) अवश्य है?
If \(A=\{1,2,3\}\), how many elements of (\mathcal{P}(A)) must contain (1)?
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#power-set
#counting
#fixed-element
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A (4)
B (3)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
Fix (1), then each of the remaining (2) elements may be chosen or not, giving \(2^2=4\) subsets. For a fixed element, take powers over the remaining elements.
Step 2
Why this answer is correct
The correct answer is A. (4). Fix (1), then each of the remaining (2) elements may be chosen or not, giving \(2^2=4\) subsets. For a fixed element, take powers over the remaining elements.
Step 3
Exam Tip
(1) को निश्चित रखकर बाकी (2) तत्वों को चुनने या न चुनने के \(2^2=4\) तरीके हैं। निश्चित तत्व वाले प्रश्न में बाकी तत्वों पर घात लगाएं।
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यदि \(A=\{p,q,r,s\}\), तो (\mathcal{P}(A)) में (p) नहीं रखने वाले कितने तत्व होंगे?
If \(A=\{p,q,r,s\}\), how many elements of (\mathcal{P}(A)) do not contain (p)?
#sets
#power-set
#counting
#excluded-element
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A (8)
B (4)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
Excluding (p), subsets are formed only from (q,r,s), so \(2^3=8\). When one element is forbidden, take the power of the remaining elements.
Step 2
Why this answer is correct
The correct answer is A. (8). Excluding (p), subsets are formed only from (q,r,s), so \(2^3=8\). When one element is forbidden, take the power of the remaining elements.
Step 3
Exam Tip
(p) को बाहर रखकर केवल (q,r,s) से उपसमुच्चय बनेंगे, इसलिए \(2^3=8\)। किसी तत्व को हटाने पर शेष तत्वों की शक्ति लें।
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यदि \(U={1,2,\ldots,15}\), \(A=\{3,6,9,12,15\}\), तो (n(\mathcal{P}(A'))) कितना है?
If \(U={1,2,\ldots,15}\), \(A=\{3,6,9,12,15\}\), what is (n(\mathcal{P}(A')))?
#sets
#power-set
#universal-set
#multiples
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A (1024)
B (32)
C (512)
D (32768)
Explanation opens after your attempt
Step 1
Concept
(A) has (5) elements and (U) has (15), so (A') has (10) elements. Therefore its power set has \(2^{10}=1024\) elements.
Step 2
Why this answer is correct
The correct answer is A. (1024). (A) has (5) elements and (U) has (15), so (A') has (10) elements. Therefore its power set has \(2^{10}=1024\) elements.
Step 3
Exam Tip
(A) में (5) तत्व हैं और (U) में (15), इसलिए (A') में (10) तत्व हैं। अतः घात समुच्चय में \(2^{10}=1024\) तत्व हैं।
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यदि \(A=\{1,2,{1,2}\}\), तो कौन सा कथन सही है?
If \(A=\{1,2,{1,2}\}\), which statement is correct?
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#power-set
#membership
#subset
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A \({1,2}\in A\) और \({1,2}\in\mathcal{P}(A)\) / \({1,2}\in A\) and \({1,2}\in\mathcal{P}(A)\)
B \({1,2}\notin A\) और \({1,2}\in\mathcal{P}(A)\) / \({1,2}\notin A\) and \({1,2}\in\mathcal{P}(A)\)
C \({1,2}\in A\) लेकिन \({1,2}\notin\mathcal{P}(A)\) / \({1,2}\in A\) but \({1,2}\notin\mathcal{P}(A)\)
D \(1\in\mathcal{P}(A)\)
Explanation opens after your attempt
Correct Answer
A. \({1,2}\in A\) और \({1,2}\in\mathcal{P}(A)\) / \({1,2}\in A\) and \({1,2}\in\mathcal{P}(A)\)
Step 1
Concept
({1,2}) is an element of (A) and also a subset of (A). Hence it belongs to both (A) and (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is A. \({1,2}\in A\) और \({1,2}\in\mathcal{P}(A)\) / \({1,2}\in A\) and \({1,2}\in\mathcal{P}(A)\). ({1,2}) is an element of (A) and also a subset of (A). Hence it belongs to both (A) and (\mathcal{P}(A)).
Step 3
Exam Tip
({1,2}) (A) का एक तत्व है और साथ ही (A) का उपसमुच्चय भी है। इसलिए यह (A) में भी है और (\mathcal{P}(A)) में भी है।
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यदि \(U=\{1,2,3,4,5,6\}\) और (A'={2,4,6}), तो (A) क्या है?
If \(U=\{1,2,3,4,5,6\}\) and (A'={2,4,6}), what is (A)?
#sets
#universal-set
#complement
#reverse
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A ({1,3,5})
B ({2,4,6})
C ({1,2,3,4,5,6})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({1,3,5})
Step 1
Concept
(A) contains elements that are in (U) but not in (A'). Therefore \(A=\{1,3,5\}\).
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5}). (A) contains elements that are in (U) but not in (A'). Therefore \(A=\{1,3,5\}\).
Step 3
Exam Tip
(A) वही तत्व होंगे जो (U) में हैं पर (A') में नहीं हैं। इसलिए \(A=\{1,3,5\}\)।
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यदि (n(A)=5), तो (\mathcal{P}(A)) के कितने तत्वों में कम से कम (4) तत्व होंगे?
If (n(A)=5), how many elements of (\mathcal{P}(A)) have at least (4) elements?
#sets
#power-set
#combination
#at-least
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A (6)
B (10)
C (16)
D (26)
Explanation opens after your attempt
Step 1
Concept
Subsets with at least (4) elements are (4)-element and (5)-element subsets. The count is \(\binom{5}{4}+\binom{5}{5}=5+1=6\).
Step 2
Why this answer is correct
The correct answer is A. (6). Subsets with at least (4) elements are (4)-element and (5)-element subsets. The count is \(\binom{5}{4}+\binom{5}{5}=5+1=6\).
Step 3
Exam Tip
कम से कम (4) तत्वों वाले उपसमुच्चय (4)-तत्वीय और (5)-तत्वीय होंगे। संख्या \(\binom{5}{4}+\binom{5}{5}=5+1=6\) है।
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यदि \(A=\{a,b,c\}\), तो (\mathcal{P}(A)) में ठीक विषम संख्या के तत्वों वाले कितने उपसमुच्चय हैं?
If \(A=\{a,b,c\}\), how many subsets in (\mathcal{P}(A)) have an odd number of elements?
#sets
#power-set
#odd-cardinality
#combination
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A (4)
B (3)
C (5)
D (8)
Explanation opens after your attempt
Step 1
Concept
The odd sizes are (1) and (3), so the count is \(\binom{3}{1}+\binom{3}{3}=3+1=4\). Count subsets by their sizes.
Step 2
Why this answer is correct
The correct answer is A. (4). The odd sizes are (1) and (3), so the count is \(\binom{3}{1}+\binom{3}{3}=3+1=4\). Count subsets by their sizes.
Step 3
Exam Tip
विषम आकार (1) और (3) हैं, इसलिए संख्या \(\binom{3}{1}+\binom{3}{3}=3+1=4\) है। आकार के आधार पर उपसमुच्चय गिनें।
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यदि \(A\subseteq B\), तो (\mathcal{P}(A)) और (\mathcal{P}(B)) के बीच कौन सा संबंध हमेशा सही है?
If \(A\subseteq B\), which relation between (\mathcal{P}(A)) and (\mathcal{P}(B)) is always true?
#sets
#power-set
#subset-relation
#reasoning
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A \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\)
B \(\mathcal{P}(B)\subseteq\mathcal{P}(A)\)
C (\mathcal{P}(A)=B)
D \(A\in\mathcal{P}(B)\) गलत है / \(A\in\mathcal{P}(B)\) is false
Explanation opens after your attempt
Correct Answer
A. \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\)
Step 1
Concept
Every subset of (A) is also a subset of (B). Therefore the power set relation remains in the same direction.
Step 2
Why this answer is correct
The correct answer is A. \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\). Every subset of (A) is also a subset of (B). Therefore the power set relation remains in the same direction.
Step 3
Exam Tip
(A) का हर उपसमुच्चय (B) का भी उपसमुच्चय होगा। इसलिए घात समुच्चय का संबंध भी उसी दिशा में रहता है।
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यदि \(U={1,2,\ldots,30}\), (A) (2) के गुणजों का समुच्चय और (B) (5) के गुणजों का समुच्चय है, तो (n(\(A\cup B\)')) कितना है?
If \(U={1,2,\ldots,30}\), (A) is the set of multiples of (2), and (B) is the set of multiples of (5), what is (n(\(A\cup B\)'))?
#sets
#universal-set
#complement
#inclusion-exclusion
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A (12)
B (18)
C (21)
D (9)
Explanation opens after your attempt
Step 1
Concept
There are (15) multiples of (2), (6) multiples of (5), and (3) multiples of (10), so the union has (18) elements. The complement has (30-18=12) elements.
Step 2
Why this answer is correct
The correct answer is A. (12). There are (15) multiples of (2), (6) multiples of (5), and (3) multiples of (10), so the union has (18) elements. The complement has (30-18=12) elements.
Step 3
Exam Tip
(2) के (15), (5) के (6), और (10) के (3) गुणज हैं, इसलिए संघ में (18) तत्व हैं। पूरक में (30-18=12) तत्व होंगे।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) में ऐसे उपसमुच्चय कितने हैं जिनमें (1) है पर (4) नहीं है?
If \(A=\{1,2,3,4\}\), how many subsets in (\mathcal{P}(A)) contain (1) but not (4)?
#sets
#power-set
#counting
#condition
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A (4)
B (6)
C (8)
D (2)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (4) is excluded, so (2,3) are free. Thus there are \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is A. (4). (1) is fixed and (4) is excluded, so (2,3) are free. Thus there are \(2^2=4\) subsets.
Step 3
Exam Tip
(1) निश्चित है और (4) बाहर है, इसलिए (2,3) स्वतंत्र हैं। कुल \(2^2=4\) उपसमुच्चय होंगे।
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यदि \(U=\{a,b,c,d\}\), तो (\mathcal{P}(U)) में ऐसे कितने तत्व हैं जिनका पूरक भी एक-तत्वीय है?
If \(U=\{a,b,c,d\}\), how many elements of (\mathcal{P}(U)) have a singleton complement?
#sets
#power-set
#complement
#combination
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A (4)
B (1)
C (8)
D (6)
Explanation opens after your attempt
Step 1
Concept
For the complement to be singleton, the original subset must have (3) elements. There are \(\binom{4}{3}=4\) such subsets.
Step 2
Why this answer is correct
The correct answer is A. (4). For the complement to be singleton, the original subset must have (3) elements. There are \(\binom{4}{3}=4\) such subsets.
Step 3
Exam Tip
पूरक एक-तत्वीय होने के लिए मूल उपसमुच्चय में (3) तत्व होने चाहिए। ऐसे उपसमुच्चय \(\binom{4}{3}=4\) हैं।
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यदि (A) रिक्त समुच्चय है, तो (\mathcal{P}(A)) के बारे में कौन सा कथन सही है?
If (A) is the empty set, which statement about (\mathcal{P}(A)) is correct?
#sets
#power-set
#empty-set
#definition
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A (\mathcal{P}(A)={\varnothing})
B (\mathcal{P}(A)=\varnothing)
C \(\varnothing\notin\mathcal{P}(A)\)
D (n(\mathcal{P}(A))=0)
Explanation opens after your attempt
Correct Answer
A. (\mathcal{P}(A)={\varnothing})
Step 1
Concept
The only subset of the empty set is the empty set itself. Therefore its power set is \({\varnothing}\).
Step 2
Why this answer is correct
The correct answer is A. (\mathcal{P}(A)={\varnothing}). The only subset of the empty set is the empty set itself. Therefore its power set is \({\varnothing}\).
Step 3
Exam Tip
रिक्त समुच्चय का एकमात्र उपसमुच्चय स्वयं रिक्त समुच्चय है। इसलिए उसका घात समुच्चय \({\varnothing}\) है।
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यदि \(U={1,2,\ldots,10}\) और \(A={x:x\) (2) या (3) का गुणज है(}), तो (A') क्या है?
If \(U={1,2,\ldots,10}\) and \(A={x:x\) is a multiple of (2) or (3)(}), what is (A')?
#sets
#universal-set
#complement
#multiples
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A ({1,5,7})
B ({2,3,4,6,8,9,10})
C ({5,7})
D ({1,5,7,9})
Explanation opens after your attempt
Correct Answer
A. ({1,5,7})
Step 1
Concept
\(A=\{2,3,4,6,8,9,10\}\), so the remaining numbers are ({1,5,7}). The word or means union.
Step 2
Why this answer is correct
The correct answer is A. ({1,5,7}). \(A=\{2,3,4,6,8,9,10\}\), so the remaining numbers are ({1,5,7}). The word or means union.
Step 3
Exam Tip
\(A=\{2,3,4,6,8,9,10\}\), इसलिए बची संख्याएं ({1,5,7}) हैं। या का अर्थ संघ होता है।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) में ठीक (2) या (3) तत्वों वाले कुल कितने उपसमुच्चय हैं?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) have exactly (2) or (3) elements?
#sets
#power-set
#combination
#counting
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A (20)
B (10)
C (25)
D (32)
Explanation opens after your attempt
Step 1
Concept
The count is \(\binom{5}{2}+\binom{5}{3}=10+10=20\). In such questions, add the required sizes separately.
Step 2
Why this answer is correct
The correct answer is A. (20). The count is \(\binom{5}{2}+\binom{5}{3}=10+10=20\). In such questions, add the required sizes separately.
Step 3
Exam Tip
संख्या \(\binom{5}{2}+\binom{5}{3}=10+10=20\) है। ऐसे प्रश्नों में आकारों को अलग-अलग जोड़ें।
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यदि \(A={x:x\) अंग्रेजी शब्द sets के अलग-अलग अक्षर हैं(}), तो (n(\mathcal{P}(A))) कितना है?
If \(A={x:x\) is a distinct letter of the English word sets(}), what is (n(\mathcal{P}(A)))?
#sets
#power-set
#distinct-elements
#word-problem
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A (8)
B (16)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
The word sets has distinct letters (s,e,t), so (A) has (3) elements. The power set has \(2^3=8\) elements.
Step 2
Why this answer is correct
The correct answer is A. (8). The word sets has distinct letters (s,e,t), so (A) has (3) elements. The power set has \(2^3=8\) elements.
Step 3
Exam Tip
शब्द sets में अलग-अलग अक्षर (s,e,t) हैं, इसलिए (A) में (3) तत्व हैं। घात समुच्चय में \(2^3=8\) तत्व होंगे।
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यदि \(U=\{0,1,2,3,4,5\}\) और \(A={x:x^2=x}\), तो (n(\mathcal{P}(A'))) कितना है?
If \(U=\{0,1,2,3,4,5\}\) and \(A={x:x^2=x}\), what is (n(\mathcal{P}(A')))?
#sets
#power-set
#universal-set
#equation
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A (16)
B (4)
C (8)
D (32)
Explanation opens after your attempt
Step 1
Concept
\(x^2=x\) gives (x=0) or (x=1), so \(A=\{0,1\}\). (A') has (4) elements, hence \(2^4=16\).
Step 2
Why this answer is correct
The correct answer is A. (16). \(x^2=x\) gives (x=0) or (x=1), so \(A=\{0,1\}\). (A') has (4) elements, hence \(2^4=16\).
Step 3
Exam Tip
\(x^2=x\) से (x=0) या (x=1), इसलिए \(A=\{0,1\}\)। (A') में (4) तत्व हैं, अतः \(2^4=16\)।
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यदि \(A=\{a,b,c,d,e\}\), तो (\mathcal{P}(A)) में ऐसे कितने उपसमुच्चय हैं जिनमें (a) और (b) दोनों हैं?
If \(A=\{a,b,c,d,e\}\), how many subsets in (\mathcal{P}(A)) contain both (a) and (b)?
#sets
#power-set
#counting
#fixed-elements
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A (8)
B (16)
C (10)
D (4)
Explanation opens after your attempt
Step 1
Concept
(a) and (b) are fixed, and the remaining (3) elements are free. Therefore there are \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is A. (8). (a) and (b) are fixed, and the remaining (3) elements are free. Therefore there are \(2^3=8\) subsets.
Step 3
Exam Tip
(a) और (b) निश्चित हैं, बाकी (3) तत्व स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(U={1,2,\ldots,25}\) और \(A={x:x\) पूर्ण वर्ग है(}), तो (n(A')) कितना है?
If \(U={1,2,\ldots,25}\) and \(A={x:x\) is a perfect square(}), what is (n(A'))?
#sets
#universal-set
#complement
#perfect-squares
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A (20)
B (5)
C (21)
D (19)
Explanation opens after your attempt
Step 1
Concept
The perfect squares from (1) to (25) are (1,4,9,16,25), giving (5) elements. Thus (n(A')=25-5=20).
Step 2
Why this answer is correct
The correct answer is A. (20). The perfect squares from (1) to (25) are (1,4,9,16,25), giving (5) elements. Thus (n(A')=25-5=20).
Step 3
Exam Tip
(1) से (25) तक पूर्ण वर्ग (1,4,9,16,25) हैं, यानी (5) तत्व। इसलिए (n(A')=25-5=20)।
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यदि \(A=\{1,2,3\}\), तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जो (A) के उचित उपसमुच्चय हैं?
If \(A=\{1,2,3\}\), how many elements of (\mathcal{P}(A)) are proper subsets of (A)?
#sets
#power-set
#proper-subset
#counting
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A (7)
B (8)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
A set with (3) elements has \(2^3=8\) subsets. A proper subset excludes (A) itself, so there are (7).
Step 2
Why this answer is correct
The correct answer is A. (7). A set with (3) elements has \(2^3=8\) subsets. A proper subset excludes (A) itself, so there are (7).
Step 3
Exam Tip
(3) तत्वों वाले समुच्चय के कुल \(2^3=8\) उपसमुच्चय हैं। उचित उपसमुच्चय में स्वयं (A) नहीं आता, इसलिए (7) हैं।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) में ऐसे कितने उपसमुच्चय हैं जिनमें (2) है और आकार सम है?
If \(A=\{1,2,3,4\}\), how many subsets in (\mathcal{P}(A)) contain (2) and have even size?
#sets
#power-set
#even-cardinality
#condition
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A (4)
B (8)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
After fixing (2), choose an odd number of elements from the remaining (3). The count is \(\binom{3}{1}+\binom{3}{3}=4\).
Step 2
Why this answer is correct
The correct answer is A. (4). After fixing (2), choose an odd number of elements from the remaining (3). The count is \(\binom{3}{1}+\binom{3}{3}=4\).
Step 3
Exam Tip
(2) को निश्चित रखने पर बाकी (3) से विषम संख्या तत्व चुनने होंगे। संख्या \(\binom{3}{1}+\binom{3}{3}=4\) है।
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यदि \(U={1,2,\ldots,9}\), \(A=\{1,2,3,4\}\) और \(B=\{3,4,5,6\}\), तो (\(A\cap B\)') क्या है?
If \(U={1,2,\ldots,9}\), \(A=\{1,2,3,4\}\), and \(B=\{3,4,5,6\}\), what is (\(A\cap B\)')?
#sets
#universal-set
#intersection
#complement
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A ({1,2,5,6,7,8,9})
B ({3,4})
C ({7,8,9})
D ({1,2,3,4,5,6})
Explanation opens after your attempt
Correct Answer
A. ({1,2,5,6,7,8,9})
Step 1
Concept
\(A\cap B={3,4}\), so its complement is all other elements in (U). For complement of an intersection, first find the common elements.
Step 2
Why this answer is correct
The correct answer is A. ({1,2,5,6,7,8,9}). \(A\cap B={3,4}\), so its complement is all other elements in (U). For complement of an intersection, first find the common elements.
Step 3
Exam Tip
\(A\cap B={3,4}\), इसलिए इसका पूरक (U) में बाकी सभी तत्व हैं। प्रतिच्छेद का पूरक लेते समय पहले साझा तत्व निकालें।
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यदि \(U={x:x\in\mathbb{Z}, -3\le x\le3}\) और \(A={x:x\ge0}\), तो (A') क्या है?
If \(U={x:x\in\mathbb{Z}, -3\le x\le3}\) and \(A={x:x\ge0}\), what is (A')?
#sets
#universal-set
#complement
#integers
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A ({-3,-2,-1})
B ({0,1,2,3})
C ({-3,-2,-1,0})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
A. ({-3,-2,-1})
Step 1
Concept
(U) has integers from (-3) to (3), and (A) has (0,1,2,3). Thus the complement is ({-3,-2,-1}).
Step 2
Why this answer is correct
The correct answer is A. ({-3,-2,-1}). (U) has integers from (-3) to (3), and (A) has (0,1,2,3). Thus the complement is ({-3,-2,-1}).
Step 3
Exam Tip
(U) में (-3) से (3) तक पूर्णांक हैं और (A) में (0,1,2,3) हैं। इसलिए पूरक ({-3,-2,-1}) है।
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यदि \(A=\{a,b,c\}\), तो (\mathcal{P}(A)) के कितने तत्व (A) से असंयुक्त हैं?
If \(A=\{a,b,c\}\), how many elements of (\mathcal{P}(A)) are disjoint from (A)?
#sets
#power-set
#disjoint
#empty-set
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A (1)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
Among subsets of (A), only \(\varnothing\) is disjoint from (A). Disjoint means having no common element.
Step 2
Why this answer is correct
The correct answer is A. (1). Among subsets of (A), only \(\varnothing\) is disjoint from (A). Disjoint means having no common element.
Step 3
Exam Tip
(A) के उपसमुच्चयों में केवल \(\varnothing\) ही (A) से असंयुक्त है। असंयुक्त का अर्थ साझा तत्व न होना है।
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यदि \(A=\{1,2,3,4,5,6\}\), तो (\mathcal{P}(A)) में ऐसे कितने उपसमुच्चय हैं जिनमें (1) या (2) में से कम से कम एक है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) contain at least one of (1) or (2)?
#sets
#power-set
#counting
#at-least-one
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A (48)
B (32)
C (16)
D (64)
Explanation opens after your attempt
Step 1
Concept
There are (64) total subsets, and those containing neither (1) nor (2) are \(2^4=16\). Hence the answer is (64-16=48).
Step 2
Why this answer is correct
The correct answer is A. (48). There are (64) total subsets, and those containing neither (1) nor (2) are \(2^4=16\). Hence the answer is (64-16=48).
Step 3
Exam Tip
कुल उपसमुच्चय (64) हैं और जिनमें (1,2) दोनों नहीं हैं वे \(2^4=16\) हैं। इसलिए उत्तर (64-16=48) है।
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यदि \(U=\{1,2,3,4,5\}\) और \(A=\{1,2\}\), तो \(\mathcal{P}(A')\subseteq\mathcal{P}(U)\) क्यों सत्य है?
If \(U=\{1,2,3,4,5\}\) and \(A=\{1,2\}\), why is \(\mathcal{P}(A')\subseteq\mathcal{P}(U)\) true?
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#power-set
#universal-set
#subset-relation
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A क्योंकि \(A'\subseteq U\) / Because \(A'\subseteq U\)
B क्योंकि \(U\subseteq A'\) / Because \(U\subseteq A'\)
C क्योंकि (A'=A) / Because (A'=A)
D क्योंकि (\mathcal{P}(A')=U) / Because (\mathcal{P}(A')=U)
Explanation opens after your attempt
Correct Answer
A. क्योंकि \(A'\subseteq U\) / Because \(A'\subseteq U\)
Step 1
Concept
Every complement (A') contains elements only from (U). Thus \(A'\subseteq U\) gives \(\mathcal{P}(A')\subseteq\mathcal{P}(U)\).
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \(A'\subseteq U\) / Because \(A'\subseteq U\). Every complement (A') contains elements only from (U). Thus \(A'\subseteq U\) gives \(\mathcal{P}(A')\subseteq\mathcal{P}(U)\).
Step 3
Exam Tip
किसी भी पूरक (A') के सभी तत्व (U) के अंदर ही होते हैं। इसलिए \(A'\subseteq U\) से \(\mathcal{P}(A')\subseteq\mathcal{P}(U)\) मिलता है।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) में ऐसे कितने उपसमुच्चय हैं जिनका पूरक (A) के सापेक्ष स्वयं के बराबर है?
If \(A=\{1,2,3,4\}\), how many subsets in (\mathcal{P}(A)) are equal to their own complement with respect to (A)?
#sets
#power-set
#complement
#reasoning
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A (0)
B (1)
C (2)
D (6)
Explanation opens after your attempt
Step 1
Concept
No set can be equal to its complement because each element would have to be both included and not included. Therefore the count is (0).
Step 2
Why this answer is correct
The correct answer is A. (0). No set can be equal to its complement because each element would have to be both included and not included. Therefore the count is (0).
Step 3
Exam Tip
कोई समुच्चय अपने पूरक के बराबर नहीं हो सकता क्योंकि तब वह अपने ही प्रत्येक तत्व को रखेगा और नहीं भी रखेगा। इसलिए संख्या (0) है।
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यदि \(U={1,2,\ldots,18}\) और \(A={x:x\) (18) का अभाज्य गुणनखंड है(}), तो (n(\mathcal{P}(A'))) कितना है?
If \(U={1,2,\ldots,18}\) and \(A={x:x\) is a prime factor of (18)(}), what is (n(\mathcal{P}(A')))?
#sets
#power-set
#universal-set
#prime-factors
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A (65536)
B (262144)
C (4096)
D (131072)
Explanation opens after your attempt
Correct Answer
A. (65536)
Step 1
Concept
The prime factors of (18) are (2) and (3), so (A) has (2) elements. (A') has (16) elements and \(2^{16}=65536\).
Step 2
Why this answer is correct
The correct answer is A. (65536). The prime factors of (18) are (2) and (3), so (A) has (2) elements. (A') has (16) elements and \(2^{16}=65536\).
Step 3
Exam Tip
(18) के अभाज्य गुणनखंड (2) और (3) हैं, इसलिए (A) में (2) तत्व हैं। (A') में (16) तत्व होंगे और \(2^{16}=65536\)।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) में ऐसे कितने उपसमुच्चय हैं जिनमें (1) है या (5) नहीं है?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) contain (1) or do not contain (5)?
#sets
#power-set
#counting
#or-condition
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A (24)
B (16)
C (20)
D (28)
Explanation opens after your attempt
Step 1
Concept
The opposite condition is not containing (1) and containing (5), which gives \(2^3=8\) subsets. Subtracting from (32) gives (24).
Step 2
Why this answer is correct
The correct answer is A. (24). The opposite condition is not containing (1) and containing (5), which gives \(2^3=8\) subsets. Subtracting from (32) gives (24).
Step 3
Exam Tip
पूरक स्थिति है कि (1) नहीं है और (5) है, जिसके \(2^3=8\) उपसमुच्चय हैं। कुल (32) में से (8) घटाने पर (24) मिलते हैं।
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यदि \(U={1,2,\ldots,14}\), \(A=\{2,4,6,8,10,12,14\}\) और \(B=\{1,3,5,7,9,11,13\}\), तो (\mathcal{P}(A')=\mathcal{P}(B)) के बारे में क्या सही है?
If \(U={1,2,\ldots,14}\), \(A=\{2,4,6,8,10,12,14\}\), and \(B=\{1,3,5,7,9,11,13\}\), what is true about (\mathcal{P}(A')=\mathcal{P}(B))?
#sets
#power-set
#universal-set
#equality
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A यह सत्य है क्योंकि (A'=B) / It is true because (A'=B)
B यह असत्य है क्योंकि \(A'\ne B\) / It is false because \(A'\ne B\)
C यह तभी सत्य है जब (U=A) / It is true only when (U=A)
D यह तभी सत्य है जब \(B=\varnothing\) / It is true only when \(B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. यह सत्य है क्योंकि (A'=B) / It is true because (A'=B)
Step 1
Concept
(A) is the set of even numbers, so (A') in (U) is the set of odd numbers, which is (B). Equal original sets have equal power sets.
Step 2
Why this answer is correct
The correct answer is A. यह सत्य है क्योंकि (A'=B) / It is true because (A'=B). (A) is the set of even numbers, so (A') in (U) is the set of odd numbers, which is (B). Equal original sets have equal power sets.
Step 3
Exam Tip
(A) सम संख्याओं का समुच्चय है, इसलिए (A') (U) में विषम संख्याएं हैं, जो (B) है। समान मूल समुच्चय के घात समुच्चय भी समान होते हैं।
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