यदि \(A={x:x^2-7x+12=0}\) और \(B=\{3,4\}\) हैं, तो सही संबंध कौन सा है?
If \(A={x:x^2-7x+12=0}\) and \(B=\{3,4\}\), which relation is correct?
#sets
#equal-sets
#quadratic
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Step 1
Concept
The roots are (3) and (4), so both sets have the same elements. First solve, then compare the sets.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The roots are (3) and (4), so both sets have the same elements. First solve, then compare the sets.
Step 3
Exam Tip
समीकरण के मूल (3) और (4) हैं, इसलिए दोनों समुच्चयों के सदस्य समान हैं। पहले हल निकालकर फिर समुच्चय मिलाएं।
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यदि \(A={x:x\in\mathbb{N},x\) संख्या (18) का भाजक है(}) और \(B=\{1,2,3,6,9,18\}\) हैं, तो सही संबंध क्या है?
If \(A={x:x\in\mathbb{N},x\) is a divisor of (18)(}) and \(B=\{1,2,3,6,9,18\}\), what is the correct relation?
#sets
#equal-sets
#divisors
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Step 1
Concept
The positive divisors of (18) are (1,2,3,6,9,18). Converting to roster form is the best way to compare elements.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive divisors of (18) are (1,2,3,6,9,18). Converting to roster form is the best way to compare elements.
Step 3
Exam Tip
(18) के धनात्मक भाजक (1,2,3,6,9,18) हैं। सूची रूप में बदलकर सदस्य मिलाना सबसे अच्छा तरीका है।
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यदि \(A={x:x\in\mathbb{Z},-3<x\le2}\), तो कौन सा समुच्चय (A) के बराबर है?
If \(A={x:x\in\mathbb{Z},-3<x\le2}\), which set is equal to (A)?
#sets
#integers
#equal-sets
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A ({-2,-1,0,1,2})
B ({-3,-2,-1,0,1,2})
C ({-2,-1,0,1})
D ({-3,-2,-1,0,1})
Explanation opens after your attempt
Correct Answer
A. ({-2,-1,0,1,2})
Step 1
Concept
The condition excludes (-3) and includes (2). Read open and closed endpoints carefully.
Step 2
Why this answer is correct
The correct answer is A. ({-2,-1,0,1,2}). The condition excludes (-3) and includes (2). Read open and closed endpoints carefully.
Step 3
Exam Tip
शर्त में (-3) शामिल नहीं है और (2) शामिल है। अंतराल में खुले और बंद चिह्न ध्यान से पढ़ें।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (2) को शामिल करने और (4) को शामिल न करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets contain (2) and do not contain (4)?
#subsets
#counting
#conditions
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
Element (2) is fixed, (4) is excluded, and (1,3,5) are free. Hence \(2^3=8\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (8). Element (2) is fixed, (4) is excluded, and (1,3,5) are free. Hence \(2^3=8\) subsets are possible.
Step 3
Exam Tip
(2) निश्चित है, (4) बाहर है और (1,3,5) स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय बनेंगे।
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यदि ({a+2}={9}), तो (a) का मान क्या है?
If ({a+2}={9}), what is the value of (a)?
#equal-sets
#singleton
#parameter
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A (5)
B (6)
C (7)
D (11)
Explanation opens after your attempt
Step 1
Concept
Singleton sets are equal, so (a+2=9). Hence (a=7).
Step 2
Why this answer is correct
The correct answer is C. (7). Singleton sets are equal, so (a+2=9). Hence (a=7).
Step 3
Exam Tip
एक-सदस्यीय समुच्चय बराबर हैं, इसलिए (a+2=9)। इससे (a=7) मिलता है।
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यदि ({2,a}={5,2}), तो (a) का मान क्या होगा?
If ({2,a}={5,2}), what is the value of (a)?
#equal-sets
#unordered-set
#parameter
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A (2)
B (3)
C (5)
D (7)
Explanation opens after your attempt
Step 1
Concept
Both sets already share (2), so the other element must be (5). Order does not matter in sets.
Step 2
Why this answer is correct
The correct answer is C. (5). Both sets already share (2), so the other element must be (5). Order does not matter in sets.
Step 3
Exam Tip
दोनों समुच्चयों में (2) पहले से समान है, इसलिए दूसरा सदस्य (5) होना चाहिए। समुच्चय में क्रम का महत्व नहीं होता।
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यदि \(A=\{1,1,2,2,3\}\) और \(B=\{3,2,1\}\), तो कौन सा कथन सत्य है?
If \(A=\{1,1,2,2,3\}\) and \(B=\{3,2,1\}\), which statement is true?
#sets
#duplicates
#equal-sets
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A (A=B)
B \(A\subset B\) उचित / \(A\subset B\) proper
C \(B\subset A\) उचित / \(B\subset A\) proper
D \(A\ne B\)
Explanation opens after your attempt
Step 1
Concept
Repeated elements are not counted separately in a set. Both sets contain only (1,2,3).
Step 2
Why this answer is correct
The correct answer is A. (A=B). Repeated elements are not counted separately in a set. Both sets contain only (1,2,3).
Step 3
Exam Tip
दोहराए गए सदस्य समुच्चय में अलग से नहीं गिने जाते। दोनों में केवल (1,2,3) सदस्य हैं।
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यदि \(A=\{1,2\}\), \(B=\{1,2,3,4,5,6\}\), तो कितने (X) ऐसे हैं कि \(A\subseteq X\subseteq B\)?
If \(A=\{1,2\}\), \(B=\{1,2,3,4,5,6\}\), how many (X) satisfy \(A\subseteq X\subseteq B\)?
#subsets
#counting
#between-sets
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A (8)
B (12)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
Elements (1,2) are fixed and (3,4,5,6) are optional. Thus \(2^4=16\) sets are possible.
Step 2
Why this answer is correct
The correct answer is C. (16). Elements (1,2) are fixed and (3,4,5,6) are optional. Thus \(2^4=16\) sets are possible.
Step 3
Exam Tip
(1,2) निश्चित हैं और (3,4,5,6) वैकल्पिक हैं। इसलिए \(2^4=16\) समुच्चय मिलते हैं।
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यदि \(A=\{p,q\}\), \(B=\{p,q,r,s\}\), तो कितने (X) ऐसे हैं कि \(A\subset X\subseteq B\)?
If \(A=\{p,q\}\), \(B=\{p,q,r,s\}\), how many (X) satisfy \(A\subset X\subseteq B\)?
#proper-subset
#counting
#between-sets
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
Elements (r,s) are optional, but \(X\ne A\). Therefore the answer is \(2^2-1=3\).
Step 2
Why this answer is correct
The correct answer is B. (3). Elements (r,s) are optional, but \(X\ne A\). Therefore the answer is \(2^2-1=3\).
Step 3
Exam Tip
(r,s) वैकल्पिक हैं लेकिन (X=A) नहीं होना चाहिए। इसलिए \(2^2-1=3\) उत्तर है।
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यदि (A) में (5) सदस्य हैं, तो (A) के तीन-सदस्यीय उपसमुच्चयों की संख्या कितनी है?
If (A) has (5) elements, how many three-element subsets does (A) have?
#subsets
#combinations
#counting
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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 three elements is \(\binom{5}{3}=10\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose three elements is \(\binom{5}{3}=10\). Order is not counted in subsets.
Step 3
Exam Tip
तीन सदस्य चुनने की संख्या \(\binom{5}{3}=10\) है। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{1,2,3,4,5\}\), तो (1) को शामिल और (5) को बाहर रखने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets contain (1) and exclude (5)?
#subsets
#counting
#conditions
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
Element (1) is fixed, (5) is forbidden, and (2,3,4) are free. Hence \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). Element (1) is fixed, (5) is forbidden, and (2,3,4) are free. Hence \(2^3=8\).
Step 3
Exam Tip
(1) निश्चित है, (5) निषिद्ध है और (2,3,4) स्वतंत्र हैं। इसलिए \(2^3=8\) है।
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यदि \(A=\{a,b,c,d\}\), तो (a) या (b) में से कम से कम एक को शामिल करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d\}\), how many subsets contain at least one of (a) or (b)?
#subsets
#counting
#at-least-one
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A (8)
B (10)
C (12)
D (14)
Explanation opens after your attempt
Step 1
Concept
Total subsets are (16), and those excluding both (a,b) are \(2^2=4\). Hence (16-4=12).
Step 2
Why this answer is correct
The correct answer is C. (12). Total subsets are (16), and those excluding both (a,b) are \(2^2=4\). Hence (16-4=12).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और (a,b) दोनों को बाहर रखने वाले \(2^2=4\) हैं। इसलिए (16-4=12) है।
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यदि (A) में (4) सदस्य हैं, तो (\mathcal{P}(A)) में कितने सदस्य होंगे?
If (A) has (4) elements, how many elements will (\mathcal{P}(A)) have?
#power-set
#counting
#subsets
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A (8)
B (12)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
A power set contains all subsets, so the number is \(2^4=16\). Think separately about elements of (A) and elements of (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is C. (16). A power set contains all subsets, so the number is \(2^4=16\). Think separately about elements of (A) and elements of (\mathcal{P}(A)).
Step 3
Exam Tip
घात समुच्चय में सभी उपसमुच्चय होते हैं, इसलिए संख्या \(2^4=16\) है। (A) के सदस्य और (\mathcal{P}(A)) के सदस्य अलग सोचें।
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यदि (n(\mathcal{P}(A))=32), तो (n(A)) क्या है?
If (n(\mathcal{P}(A))=32), what is (n(A))?
#power-set
#cardinality
#counting
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A (4)
B (5)
C (6)
D (16)
Explanation opens after your attempt
Step 1
Concept
We have (n(\mathcal{P}(A))=2^{n(A)}). Since \(32=2^5\), (n(A)=5).
Step 2
Why this answer is correct
The correct answer is B. (5). We have (n(\mathcal{P}(A))=2^{n(A)}). Since \(32=2^5\), (n(A)=5).
Step 3
Exam Tip
(n(\mathcal{P}(A))=2^{n(A)}) होता है। \(32=2^5\), इसलिए (n(A)=5) है।
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यदि \(A=\{0,{0}\}\), तो कौन सा (\mathcal{P}(A)) का सदस्य है?
If \(A=\{0,{0}\}\), which is an element of (\mathcal{P}(A))?
#power-set
#nested-set
#membership
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A ({0})
B (0)
C ({{1}})
D (1)
Explanation opens after your attempt
Step 1
Concept
({0}) is a subset of (A), so it is an element of (\mathcal{P}(A)). Elements of a power set are always subsets.
Step 2
Why this answer is correct
The correct answer is A. ({0}). ({0}) is a subset of (A), so it is an element of (\mathcal{P}(A)). Elements of a power set are always subsets.
Step 3
Exam Tip
({0}) समुच्चय (A) का उपसमुच्चय है, इसलिए यह (\mathcal{P}(A)) का सदस्य है। घात समुच्चय के सदस्य हमेशा उपसमुच्चय होते हैं।
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यदि \(A={\emptyset,2}\), तो कौन सा (A) का सदस्य भी है और (A) का उपसमुच्चय भी है?
If \(A={\emptyset,2}\), which is both an element of (A) and a subset of (A)?
#empty-set
#membership
#subsets
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A \(\emptyset\)
B (2)
C ({2})
D \({\emptyset,2}\)
Explanation opens after your attempt
Correct Answer
A. \(\emptyset\)
Step 1
Concept
\(\emptyset\) is an element of (A) and also a subset of every set. Check both membership and subset relation.
Step 2
Why this answer is correct
The correct answer is A. \(\emptyset\). \(\emptyset\) is an element of (A) and also a subset of every set. Check both membership and subset relation.
Step 3
Exam Tip
\(\emptyset\) (A) का सदस्य है और हर समुच्चय का उपसमुच्चय भी है। सदस्यता और उपसमुच्चय दोनों जांचें।
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यदि \(A={\emptyset,{\emptyset},3}\), तो कौन सा (A) का उपसमुच्चय है लेकिन (A) का सदस्य नहीं है?
If \(A={\emptyset,{\emptyset},3}\), which is a subset of (A) but not an element of (A)?
#nested-set
#subsets
#membership
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A ({3})
B \(\emptyset\)
C \({\emptyset}\)
D (3)
Explanation opens after your attempt
Step 1
Concept
({3}) is a subset because \(3\in A\). But ({3}) is not listed as an element of (A).
Step 2
Why this answer is correct
The correct answer is A. ({3}). ({3}) is a subset because \(3\in A\). But ({3}) is not listed as an element of (A).
Step 3
Exam Tip
({3}) का सदस्य (3) (A) में है, इसलिए यह उपसमुच्चय है। लेकिन ({3}) (A) में सदस्य के रूप में नहीं दिया है।
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यदि \(\emptyset\in A\) है, तो कौन सा निष्कर्ष हमेशा सही है?
If \(\emptyset\in A\), which conclusion is always correct?
#empty-set
#membership
#concept
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A (A) रिक्त नहीं है / (A) is non-empty
B \(\emptyset=A\)
C \(A\subseteq\emptyset\)
D (A) का कोई उपसमुच्चय नहीं है / (A) has no subset
Explanation opens after your attempt
Correct Answer
A. (A) रिक्त नहीं है / (A) is non-empty
Step 1
Concept
If \(\emptyset\) is an element, then (A) has at least one element. Do not confuse it with \(\emptyset\subseteq A\).
Step 2
Why this answer is correct
The correct answer is A. (A) रिक्त नहीं है / (A) is non-empty. If \(\emptyset\) is an element, then (A) has at least one element. Do not confuse it with \(\emptyset\subseteq A\).
Step 3
Exam Tip
यदि \(\emptyset\) सदस्य है, तो (A) में कम से कम एक सदस्य है। इसे \(\emptyset\subseteq A\) से न मिलाएं।
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यदि \(A\subseteq B\) और \(B\subseteq C\), तो \(A\cap C\) किसके बराबर होगा?
If \(A\subseteq B\) and \(B\subseteq C\), what is \(A\cap C\) equal to?
#subsets
#intersection
#transitivity
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A (A)
B (B)
C (C)
D \(\emptyset\)
Explanation opens after your attempt
Step 1
Concept
The conditions imply \(A\subseteq C\). When (A) lies inside (C), the intersection is (A).
Step 2
Why this answer is correct
The correct answer is A. (A). The conditions imply \(A\subseteq C\). When (A) lies inside (C), the intersection is (A).
Step 3
Exam Tip
इन शर्तों से \(A\subseteq C\) है। जब (A) (C) के अंदर हो, तो प्रतिच्छेद (A) होता है।
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यदि \(A\subseteq B\), तो (A-B) किसके बराबर है?
If \(A\subseteq B\), what is (A-B) equal to?
#subsets
#difference
#identity
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A \(\emptyset\)
B (A)
C (B)
D (B-A)
Explanation opens after your attempt
Correct Answer
A. \(\emptyset\)
Step 1
Concept
No element of (A) lies outside (B). Therefore (A-B) is the empty set.
Step 2
Why this answer is correct
The correct answer is A. \(\emptyset\). No element of (A) lies outside (B). Therefore (A-B) is the empty set.
Step 3
Exam Tip
(A) का कोई भी सदस्य (B) से बाहर नहीं है। इसलिए (A-B) रिक्त समुच्चय है।
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यदि \(A\cup B=A\), तो कौन सा निष्कर्ष हमेशा सही है?
If \(A\cup B=A\), which conclusion is always true?
#subsets
#union
#reverse-identity
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A \(B\subseteq A\)
B \(A\subseteq B\)
C \(A=B^c\)
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(B\subseteq A\)
Step 1
Concept
The union remains (A), so (B) adds no element outside (A). Hence \(B\subseteq A\).
Step 2
Why this answer is correct
The correct answer is A. \(B\subseteq A\). The union remains (A), so (B) adds no element outside (A). Hence \(B\subseteq A\).
Step 3
Exam Tip
मिलन (A) ही रहा, इसलिए (B) ने (A) के बाहर कोई सदस्य नहीं जोड़ा। अतः \(B\subseteq A\) है।
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यदि \(A\cap B=B\), तो कौन सा कथन सही है?
If \(A\cap B=B\), which statement is correct?
#subsets
#intersection
#reverse-identity
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A \(B\subseteq A\)
B \(A\subseteq B\)
C \(A=B^c\)
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(B\subseteq A\)
Step 1
Concept
The common part is the whole of (B), so every element of (B) is in (A). This means \(B\subseteq A\).
Step 2
Why this answer is correct
The correct answer is A. \(B\subseteq A\). The common part is the whole of (B), so every element of (B) is in (A). This means \(B\subseteq A\).
Step 3
Exam Tip
साझा भाग पूरा (B) है, इसलिए (B) का हर सदस्य (A) में है। यही \(B\subseteq A\) है।
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यदि (A-B=A), तो (A) और (B) के बारे में कौन सा निष्कर्ष सही है?
If (A-B=A), what conclusion about (A) and (B) is correct?
#sets
#difference
#disjoint
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A \(A\cap B=\emptyset\)
B \(A\subseteq B\)
C \(B\subseteq A\)
D (A=B)
Explanation opens after your attempt
Correct Answer
A. \(A\cap B=\emptyset\)
Step 1
Concept
Removing (B) from (A) still leaves (A), so there was no common element. Hence they are disjoint.
Step 2
Why this answer is correct
The correct answer is A. \(A\cap B=\emptyset\). Removing (B) from (A) still leaves (A), so there was no common element. Hence they are disjoint.
Step 3
Exam Tip
(A) से (B) हटाने पर (A) ही बचा, यानी कोई साझा सदस्य नहीं था। इसलिए दोनों असंबद्ध हैं।
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यदि \(A\subseteq B\), तो (\mathcal{P}(A)) और (\mathcal{P}(B)) के बीच सही संबंध क्या है?
If \(A\subseteq B\), what is the correct relation between (\mathcal{P}(A)) and (\mathcal{P}(B))?
#power-set
#subsets
#inclusion
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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 कोई संबंध नहीं / No relation
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). So the same direction holds for power sets.
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). So the same direction holds for power sets.
Step 3
Exam Tip
(A) का हर उपसमुच्चय (B) का भी उपसमुच्चय होगा। इसलिए घात समुच्चय में भी यही दिशा रहती है।
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यदि \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\), तो कौन सा निष्कर्ष सही है?
If \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\), which conclusion is correct?
#power-set
#subsets
#reasoning
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A \(A\subseteq B\)
B \(B\subseteq A\)
C \(A\cap B=\emptyset\)
D \(A=B^c\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq B\)
Step 1
Concept
The set (A) itself is an element of (\mathcal{P}(A)), so it is also in (\mathcal{P}(B)). That means \(A\subseteq B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\). The set (A) itself is an element of (\mathcal{P}(A)), so it is also in (\mathcal{P}(B)). That means \(A\subseteq B\).
Step 3
Exam Tip
(A) स्वयं (\mathcal{P}(A)) का सदस्य है, इसलिए (A) (\mathcal{P}(B)) में भी होगा। इसका अर्थ \(A\subseteq B\) है।
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यदि (A) में (2) सदस्य हैं और (B) में (3) सदस्य हैं तथा \(A\subset B\), तो (\mathcal{P}(B)-\mathcal{P}(A)) में कितने सदस्य होंगे?
If (A) has (2) elements and (B) has (3) elements with \(A\subset B\), how many elements are in (\mathcal{P}(B)-\mathcal{P}(A))?
#power-set
#difference
#counting
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(B)) has (8) elements and (\mathcal{P}(A)) has (4). The difference contains (8-4=4) elements.
Step 2
Why this answer is correct
The correct answer is B. (4). (\mathcal{P}(B)) has (8) elements and (\mathcal{P}(A)) has (4). The difference contains (8-4=4) elements.
Step 3
Exam Tip
(\mathcal{P}(B)) में (8) और (\mathcal{P}(A)) में (4) सदस्य हैं। अंतर में (8-4=4) सदस्य होंगे।
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यदि \(A=\{1,2,3,4\}\), तो केवल सम सदस्यों से बने उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets can be formed using only even elements?
#subsets
#counting
#even-elements
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
The even elements are (2,4). The number of their subsets is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is C. (4). The even elements are (2,4). The number of their subsets is \(2^2=4\).
Step 3
Exam Tip
सम सदस्य (2,4) हैं। इनके सभी उपसमुच्चयों की संख्या \(2^2=4\) है।
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यदि \(A=\{1,2,3,4,5,6\}\), तो ठीक दो विषम सदस्यों वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets have exactly two odd elements?
#subsets
#counting
#odd-elements
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A (3)
B (6)
C (12)
D (24)
Explanation opens after your attempt
Step 1
Concept
Choose two from the three odd numbers, and the three even numbers are optional. The count is \(\binom{3}{2}\times2^3=24\).
Step 2
Why this answer is correct
The correct answer is D. (24). Choose two from the three odd numbers, and the three even numbers are optional. The count is \(\binom{3}{2}\times2^3=24\).
Step 3
Exam Tip
तीन विषम संख्याओं में से दो चुनें और तीन सम संख्याएँ वैकल्पिक हैं। संख्या \(\binom{3}{2}\times2^3=24\) है।
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यदि \(A=\{1,2,3,4,5\}\), तो कम से कम (4) सदस्य वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets have at least (4) elements?
#subsets
#counting
#at-least
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A (5)
B (6)
C (10)
D (16)
Explanation opens after your attempt
Step 1
Concept
There are \(\binom{5}{4}=5\) four-element subsets and (1) five-element subset. Total is (5+1=6).
Step 2
Why this answer is correct
The correct answer is B. (6). There are \(\binom{5}{4}=5\) four-element subsets and (1) five-element subset. Total is (5+1=6).
Step 3
Exam Tip
चार-सदस्यीय उपसमुच्चय \(\binom{5}{4}=5\) और पांच-सदस्यीय (1) है। कुल (5+1=6) है।
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यदि \(A=\{a,b,c,d,e\}\), तो (a) को शामिल न करने वाले उचित उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d,e\}\), how many proper subsets do not contain (a)?
#proper-subset
#counting
#excluded-element
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A (8)
B (15)
C (16)
D (31)
Explanation opens after your attempt
Step 1
Concept
After excluding (a), (4) elements remain, giving \(2^4=16\) subsets. All of them are proper subsets of (A).
Step 2
Why this answer is correct
The correct answer is C. (16). After excluding (a), (4) elements remain, giving \(2^4=16\) subsets. All of them are proper subsets of (A).
Step 3
Exam Tip
(a) को हटाकर (4) सदस्य बचते हैं, जिनके \(2^4=16\) उपसमुच्चय हैं। ये सभी (A) के उचित उपसमुच्चय हैं।
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यदि \(A=\{1,2,3,4\}\), तो (2) और (3) को साथ-साथ शामिल या साथ-साथ बाहर रखने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets either contain both (2,3) or contain neither of them?
#subsets
#counting
#paired-condition
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A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are two valid states for (2,3), and (1,4) are free. Hence \(2\times2^2=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). There are two valid states for (2,3), and (1,4) are free. Hence \(2\times2^2=8\).
Step 3
Exam Tip
(2,3) के लिए दो वैध स्थितियां हैं और (1,4) स्वतंत्र हैं। इसलिए \(2\times2^2=8\) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4\}\), तो कौन सा समुच्चय \(A\cap B\) के बराबर है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4\}\), which set is equal to \(A\cap B\)?
#sets
#intersection
#equal-sets
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A ({2,3})
B ({1,4})
C ({1,2,3,4})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({2,3})
Step 1
Concept
Intersection contains only elements common to both sets. Here the common elements are (2) and (3).
Step 2
Why this answer is correct
The correct answer is A. ({2,3}). Intersection contains only elements common to both sets. Here the common elements are (2) and (3).
Step 3
Exam Tip
प्रतिच्छेद में केवल वे सदस्य आते हैं जो दोनों में हों। यहाँ साझा सदस्य (2) और (3) हैं।
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यदि \(A\subseteq B\) और \(B-A=\emptyset\), तो कौन सा निष्कर्ष सही है?
If \(A\subseteq B\) and \(B-A=\emptyset\), which conclusion is correct?
#equal-sets
#difference
#logic
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A (A=B)
B \(A\subset B\) उचित / \(A\subset B\) proper
C \(B\subset A\) उचित / \(B\subset A\) proper
D \(A=\emptyset\) जरूरी / \(A=\emptyset\) necessarily
Explanation opens after your attempt
Step 1
Concept
We have \(A\subseteq B\), and (B) has no element outside (A). Therefore the two sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). We have \(A\subseteq B\), and (B) has no element outside (A). Therefore the two sets are equal.
Step 3
Exam Tip
\(A\subseteq B\) है और (B) में (A) के बाहर कोई सदस्य नहीं है। इसलिए दोनों बराबर हैं।
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यदि \(A\triangle B={7}\) और \(A\subseteq B\), तो (B-A) किसके बराबर है?
If \(A\triangle B={7}\) and \(A\subseteq B\), what is (B-A) equal to?
#symmetric-difference
#subsets
#difference
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A ({7})
B \(\emptyset\)
C (A)
D (B)
Explanation opens after your attempt
Step 1
Concept
When \(A\subseteq B\), the symmetric difference is just (B-A). Hence (B-A={7}).
Step 2
Why this answer is correct
The correct answer is A. ({7}). When \(A\subseteq B\), the symmetric difference is just (B-A). Hence (B-A={7}).
Step 3
Exam Tip
जब \(A\subseteq B\), तब सममित अंतर (B-A) ही होता है। इसलिए (B-A={7}) है।
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यदि \(A={x:x\) (20) से छोटा (4) का धनात्मक गुणज है(}), तो कौन सा (A) के बराबर है?
If \(A={x:x\) is a positive multiple of (4) less than (20)(}), which is equal to (A)?
#equal-sets
#multiples
#set-builder
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A ({4,8,12,16})
B ({0,4,8,12,16})
C ({4,8,12,16,20})
D ({8,12,16})
Explanation opens after your attempt
Correct Answer
A. ({4,8,12,16})
Step 1
Concept
The positive multiples less than (20) are (4,8,12,16). Check both the boundary and positivity condition.
Step 2
Why this answer is correct
The correct answer is A. ({4,8,12,16}). The positive multiples less than (20) are (4,8,12,16). Check both the boundary and positivity condition.
Step 3
Exam Tip
(20) से छोटे धनात्मक गुणज (4,8,12,16) हैं। सीमा और धनात्मक शर्त दोनों जांचें।
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यदि (A) संख्या (30) के अभाज्य भाजकों का समुच्चय है, तो कौन सा (A) का उचित उपसमुच्चय है?
If (A) is the set of prime divisors of (30), which is a proper subset of (A)?
#proper-subset
#prime-divisors
#application
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A ({2,5})
B ({2,3,5})
C ({2,7})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
A. ({2,5})
Step 1
Concept
Here \(A=\{2,3,5\}\), and ({2,5}) lies inside it but is not equal to it. Hence it is a proper subset.
Step 2
Why this answer is correct
The correct answer is A. ({2,5}). Here \(A=\{2,3,5\}\), and ({2,5}) lies inside it but is not equal to it. Hence it is a proper subset.
Step 3
Exam Tip
\(A=\{2,3,5\}\) है और ({2,5}) इसके अंदर है पर बराबर नहीं है। इसलिए यह उचित उपसमुच्चय है।
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यदि \(A={x:x\in\mathbb{Z},x^2\le4}\), तो (A) के उचित उपसमुच्चयों की संख्या कितनी है?
If \(A={x:x\in\mathbb{Z},x^2\le4}\), how many proper subsets does (A) have?
#proper-subset
#counting
#integers
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A (15)
B (31)
C (32)
D (63)
Explanation opens after your attempt
Step 1
Concept
\(A=\{-2,-1,0,1,2\}\) has (5) elements. Proper subsets are \(2^5-1=31\).
Step 2
Why this answer is correct
The correct answer is B. (31). \(A=\{-2,-1,0,1,2\}\) has (5) elements. Proper subsets are \(2^5-1=31\).
Step 3
Exam Tip
\(A=\{-2,-1,0,1,2\}\) में (5) सदस्य हैं। उचित उपसमुच्चय \(2^5-1=31\) होंगे।
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यदि \(A={x:x\in\mathbb{Z},x^2=16}\) और \(B=\{-4,4\}\), तो कौन सा सही है?
If \(A={x:x\in\mathbb{Z},x^2=16}\) and \(B=\{-4,4\}\), which is correct?
#equal-sets
#integers
#solution-set
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A (A=B)
B \(A=\{4\}\)
C \(B\subset A\) उचित / \(B\subset A\) proper
D \(A=\emptyset\)
Explanation opens after your attempt
Step 1
Concept
The integer solutions are both (-4) and (4). Do not miss the negative solution in square equations.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The integer solutions are both (-4) and (4). Do not miss the negative solution in square equations.
Step 3
Exam Tip
पूर्णांक हल (-4) और (4) दोनों हैं। वर्ग वाले प्रश्नों में ऋणात्मक हल न छोड़ें।
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यदि \(A=\{1,2,3\}\), तो कौन सा (\mathcal{P}(A)) का उपसमुच्चय है?
If \(A=\{1,2,3\}\), which is a subset of (\mathcal{P}(A))?
#power-set
#subset-of-power-set
#membership
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A \({\emptyset,{1},{2,3}}\)
B ({1,{2}})
C ({4,{1}})
D ({{4}})
Explanation opens after your attempt
Correct Answer
A. \({\emptyset,{1},{2,3}}\)
Step 1
Concept
Elements of (\mathcal{P}(A)) are subsets of (A), and every member in option A is such a subset. Distinguish (1) from ({1}).
Step 2
Why this answer is correct
The correct answer is A. \({\emptyset,{1},{2,3}}\). Elements of (\mathcal{P}(A)) are subsets of (A), and every member in option A is such a subset. Distinguish (1) from ({1}).
Step 3
Exam Tip
(\mathcal{P}(A)) के सदस्य (A) के उपसमुच्चय हैं, और विकल्प A के सभी सदस्य ऐसे हैं। संख्या (1) को ({1}) से अलग समझें।
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यदि \(A=\{1,{2},3\}\), तो कौन सा कथन सत्य है?
If \(A=\{1,{2},3\}\), which statement is true?
#nested-set
#membership
#subsets
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A \({2}\in A\)
B \(2\in A\)
C \({1,2}\subseteq A\)
D \({2,3}\subseteq A\)
Explanation opens after your attempt
Correct Answer
A. \({2}\in A\)
Step 1
Concept
Here ({2}) is present as one element, but (2) itself is not an element. Brackets are very important in nested sets.
Step 2
Why this answer is correct
The correct answer is A. \({2}\in A\). Here ({2}) is present as one element, but (2) itself is not an element. Brackets are very important in nested sets.
Step 3
Exam Tip
यहाँ ({2}) एक सदस्य के रूप में मौजूद है, लेकिन (2) स्वयं सदस्य नहीं है। नेस्टेड सेट में कोष्ठक बहुत महत्वपूर्ण हैं।
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यदि \(A=\{1,2\}\), \(B=\{1,2,{1,2}\}\), तो कौन सा कथन सही है?
If \(A=\{1,2\}\), \(B=\{1,2,{1,2}\}\), which statement is correct?
#nested-set
#membership
#subset
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A \(A\in B\) और \(A\subseteq B\) दोनों / Both \(A\in B\) and \(A\subseteq B\)
B केवल \(A\in B\) / Only \(A\in B\)
C केवल \(A\subseteq B\) / Only \(A\subseteq B\)
D न तो \(A\in B\) न \(A\subseteq B\) / Neither \(A\in B\) nor \(A\subseteq B\)
Explanation opens after your attempt
Correct Answer
A. \(A\in B\) और \(A\subseteq B\) दोनों / Both \(A\in B\) and \(A\subseteq B\)
Step 1
Concept
(B) contains \(A=\{1,2\}\) as an element, and it also contains (1,2). Hence both relations are true.
Step 2
Why this answer is correct
The correct answer is A. \(A\in B\) और \(A\subseteq B\) दोनों / Both \(A\in B\) and \(A\subseteq B\). (B) contains \(A=\{1,2\}\) as an element, and it also contains (1,2). Hence both relations are true.
Step 3
Exam Tip
(B) में \(A=\{1,2\}\) सदस्य के रूप में है और (1,2) भी (B) में हैं। इसलिए दोनों संबंध सही हैं।
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यदि (A) और (B) सीमित समुच्चय हैं, \(A\subset B\), (n(A)=6), तो (n(B)) के लिए कौन सा मान असंभव है?
If (A) and (B) are finite sets, \(A\subset B\), (n(A)=6), which value of (n(B)) is impossible?
#proper-subset
#finite-sets
#cardinality
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A (6)
B (7)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
For a proper subset, the larger set must have more elements. So (n(B)=6) is impossible.
Step 2
Why this answer is correct
The correct answer is A. (6). For a proper subset, the larger set must have more elements. So (n(B)=6) is impossible.
Step 3
Exam Tip
उचित उपसमुच्चय में बड़े समुच्चय की सदस्य संख्या अधिक होनी चाहिए। इसलिए (n(B)=6) असंभव है।
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यदि \(A\subseteq B\), (n(A)=4), (n(B)=4), और दोनों सीमित हैं, तो कौन सा कथन सही है?
If \(A\subseteq B\), (n(A)=4), (n(B)=4), and both are finite, which statement is correct?
#equal-sets
#finite-sets
#cardinality
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A (A=B)
B \(A\subset B\) उचित / \(A\subset B\) proper
C (B-A) में (4) सदस्य हैं / (B-A) has (4) elements
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Step 1
Concept
For finite sets, a subset with equal cardinality leaves no extra element. Therefore the sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). For finite sets, a subset with equal cardinality leaves no extra element. Therefore the sets are equal.
Step 3
Exam Tip
सीमित सेट में उपसमुच्चय और बराबर सदस्य संख्या होने पर कोई अतिरिक्त सदस्य नहीं बचता। इसलिए सेट बराबर हैं।
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यदि \(A=\{2,4,6,8\}\) और \(B={x:x=2n,\ n\in{1,2,3,4}}\), तो कौन सा संबंध सही है?
If \(A=\{2,4,6,8\}\) and \(B={x:x=2n,\ n\in{1,2,3,4}}\), which relation is correct?
#equal-sets
#set-builder
#multiples
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A (A=B)
B \(A\subset B\) उचित / \(A\subset B\) proper
C \(B\subset A\) उचित / \(B\subset A\) proper
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Step 1
Concept
Putting (n=1,2,3,4) gives \(B=\{2,4,6,8\}\). Convert rule form into roster form.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Putting (n=1,2,3,4) gives \(B=\{2,4,6,8\}\). Convert rule form into roster form.
Step 3
Exam Tip
(n=1,2,3,4) रखने पर \(B=\{2,4,6,8\}\) मिलता है। नियम वाले सेट को सूची रूप में बदलें।
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यदि \(A={x:x\in\mathbb{N},x\) (10) का भाजक है(}), तो कौन सा (A) का उपसमुच्चय नहीं है?
If \(A={x:x\in\mathbb{N},x\) is a divisor of (10)(}), which is not a subset of (A)?
#subsets
#divisors
#not-subset
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A ({1,2})
B ({5,10})
C ({1,10})
D ({2,4})
Explanation opens after your attempt
Correct Answer
D. ({2,4})
Step 1
Concept
Here \(A=\{1,2,5,10\}\), and (4) is not its element. Check every element while testing a subset.
Step 2
Why this answer is correct
The correct answer is D. ({2,4}). Here \(A=\{1,2,5,10\}\), and (4) is not its element. Check every element while testing a subset.
Step 3
Exam Tip
\(A=\{1,2,5,10\}\) है और (4) इसका सदस्य नहीं है। उपसमुच्चय जांचते समय हर सदस्य जांचें।
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कथन: \(\emptyset\subseteq A\) हर समुच्चय (A) के लिए सत्य है। कारण: \(\emptyset\) में ऐसा कोई सदस्य नहीं जो (A) में न हो। सही विकल्प चुनिए।
Assertion: \(\emptyset\subseteq A\) is true for every set (A). Reason: There is no element in \(\emptyset\) that is not in (A). Choose the correct option.
#assertion-reason
#empty-set
#subsets
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A कथन और कारण दोनों सही हैं, कारण सही व्याख्या है / Both assertion and reason are true, and the reason explains it
B कथन सही है, कारण गलत है / Assertion is true, reason is false
C कथन गलत है, कारण सही है / Assertion is false, reason is true
D कथन और कारण दोनों गलत हैं / Both assertion and reason are false
Explanation opens after your attempt
Correct Answer
A. कथन और कारण दोनों सही हैं, कारण सही व्याख्या है / Both assertion and reason are true, and the reason explains it
Step 1
Concept
The empty set has no counterexample element. Therefore it is a subset of every set.
Step 2
Why this answer is correct
The correct answer is A. कथन और कारण दोनों सही हैं, कारण सही व्याख्या है / Both assertion and reason are true, and the reason explains it. The empty set has no counterexample element. Therefore it is a subset of every set.
Step 3
Exam Tip
रिक्त समुच्चय में कोई विरोधी सदस्य नहीं होता। इसलिए यह हर समुच्चय का उपसमुच्चय है।
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कथन: यदि (A=B), तो \(A\subseteq B\)। कारण: बराबर समुच्चयों में सभी सदस्य समान होते हैं। सही विकल्प चुनिए।
Assertion: If (A=B), then \(A\subseteq B\). Reason: Equal sets have all elements identical. Choose the correct option.
#assertion-reason
#equal-sets
#subsets
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A कथन और कारण दोनों सही हैं, कारण सही व्याख्या है / Both assertion and reason are true, and the reason explains it
B कथन सही है, कारण गलत है / Assertion is true, reason is false
C कथन गलत है, कारण सही है / Assertion is false, reason is true
D कथन और कारण दोनों गलत हैं / Both assertion and reason are false
Explanation opens after your attempt
Correct Answer
A. कथन और कारण दोनों सही हैं, कारण सही व्याख्या है / Both assertion and reason are true, and the reason explains it
Step 1
Concept
When sets are equal, every element of (A) is in (B). So the subset relation follows directly.
Step 2
Why this answer is correct
The correct answer is A. कथन और कारण दोनों सही हैं, कारण सही व्याख्या है / Both assertion and reason are true, and the reason explains it. When sets are equal, every element of (A) is in (B). So the subset relation follows directly.
Step 3
Exam Tip
बराबर समुच्चय होने पर (A) का हर सदस्य (B) में होता है। इसलिए उपसमुच्चय संबंध तुरंत मिलता है।
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कथन: यदि (n(A)=n(B)), तो (A=B)। कारण: समान सदस्य संख्या हमेशा समान सदस्य देती है। सही विकल्प चुनिए।
Assertion: If (n(A)=n(B)), then (A=B). Reason: Equal cardinality always gives identical elements. Choose the correct option.
#assertion-reason
#cardinality
#equal-sets
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A कथन और कारण दोनों सही हैं / Both assertion and reason are true
B कथन सही है, कारण गलत है / Assertion is true, reason is false
C कथन गलत है, कारण सही है / Assertion is false, reason is true
D कथन और कारण दोनों गलत हैं / Both assertion and reason are false
Explanation opens after your attempt
Correct Answer
D. कथन और कारण दोनों गलत हैं / Both assertion and reason are false
Step 1
Concept
Equal cardinality does not force identical elements. For example, ({1,2}) and ({3,4}) are not equal.
Step 2
Why this answer is correct
The correct answer is D. कथन और कारण दोनों गलत हैं / Both assertion and reason are false. Equal cardinality does not force identical elements. For example, ({1,2}) and ({3,4}) are not equal.
Step 3
Exam Tip
सदस्य संख्या समान होने से सदस्य समान होना जरूरी नहीं है। जैसे ({1,2}) और ({3,4}) बराबर नहीं हैं।
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यदि \(A=\{1,2,3,4\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें सदस्यों का योग सम हो?
If \(A=\{1,2,3,4\}\), how many subsets have an even sum of elements?
#subsets
#counting
#parity
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A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
For an even sum, the number of selected odd elements must be even. Among the (16) subsets, half, that is (8), give even sum.
Step 2
Why this answer is correct
The correct answer is C. (8). For an even sum, the number of selected odd elements must be even. Among the (16) subsets, half, that is (8), give even sum.
Step 3
Exam Tip
सम योग के लिए चुने गए विषम सदस्यों की संख्या सम होनी चाहिए। चार-सदस्यीय सेट में कुल (16) उपसमुच्चयों में आधे यानी (8) सम योग देते हैं।
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यदि \(A=\{1,2,3\}\), तो कितने क्रमित युग्म ((X,Y)) ऐसे हैं कि \(X\subseteq A\), \(Y\subseteq A\), और \(X\subseteq Y\)?
If \(A=\{1,2,3\}\), how many ordered pairs ((X,Y)) satisfy \(X\subseteq A\), \(Y\subseteq A\), and \(X\subseteq Y\)?
#subsets
#ordered-pairs
#counting
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A (8)
B (18)
C (27)
D (36)
Explanation opens after your attempt
Step 1
Concept
Each element has three choices: in neither set, in (Y) only, or in both. Hence there are \(3^3=27\) pairs.
Step 2
Why this answer is correct
The correct answer is C. (27). Each element has three choices: in neither set, in (Y) only, or in both. Hence there are \(3^3=27\) pairs.
Step 3
Exam Tip
हर सदस्य के लिए तीन विकल्प हैं: किसी में नहीं, केवल (Y) में, या दोनों में। इसलिए कुल \(3^3=27\) युग्म हैं।
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