यदि \(A=\{2,4,6\}\) और \(B=\{6,2,4,4\}\) हैं, तो सही निष्कर्ष क्या है?
If \(A=\{2,4,6\}\) and \(B=\{6,2,4,4\}\), what is the correct conclusion?
#sets
#equal-sets
#duplicates
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A बराबर / Equal
B केवल उपसमुच्चय / Subset only
C असंबंधित / Unrelated
D सर्वसमुच्चय / Universal set
Explanation opens after your attempt
Correct Answer
A. बराबर / Equal
Step 1
Concept
Both sets have exactly the members (2,4,6). Order and repetition do not matter in a set.
Step 2
Why this answer is correct
The correct answer is A. बराबर / Equal. Both sets have exactly the members (2,4,6). Order and repetition do not matter in a set.
Step 3
Exam Tip
दोनों में वास्तविक सदस्य (2,4,6) ही हैं। सेट में क्रम और दोहराव नहीं गिने जाते।
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यदि \(A={\emptyset,{\emptyset},1}\) है, तो कौन सा समुच्चय (\mathcal{P}(A)) का तत्व है लेकिन (A) का तत्व नहीं है?
If \(A={\emptyset,{\emptyset},1}\), which set is an element of (\mathcal{P}(A)) but not an element of (A)?
#sets
#power-set
#subset-membership
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A (1)
B \({\emptyset,1}\)
C \(\emptyset\)
D \({\emptyset}\)
Explanation opens after your attempt
Correct Answer
B. \({\emptyset,1}\)
Step 1
Concept
The set \({\emptyset,1}\) is a subset of (A), so it is an element of (\mathcal{P}(A)). But it is not listed as an element of (A).
Step 2
Why this answer is correct
The correct answer is B. \({\emptyset,1}\). The set \({\emptyset,1}\) is a subset of (A), so it is an element of (\mathcal{P}(A)). But it is not listed as an element of (A).
Step 3
Exam Tip
\({\emptyset,1}\), (A) का उपसमुच्चय है, इसलिए यह (\mathcal{P}(A)) का तत्व है। लेकिन यह (A) में सदस्य के रूप में नहीं दिया है।
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यदि (A) संख्या (12) के (5) से छोटे धनात्मक भाजकों का समुच्चय है और \(B=\{1,2,3,4\}\) है, तो कौन सा कथन सही है?
If (A) is the set of positive divisors of (12) that are less than (5) and \(B=\{1,2,3,4\}\), which statement is correct?
#sets
#set-builder
#equal-sets
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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 (12) less than (5) are (1,2,3,4). Sets are equal when all members match.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive divisors of (12) less than (5) are (1,2,3,4). Sets are equal when all members match.
Step 3
Exam Tip
(12) के (5) से छोटे धनात्मक भाजक (1,2,3,4) हैं। सदस्य समान हों तो सेट बराबर होते हैं।
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यदि \(A=\{1,3,5\}\) और (B) संख्या (6) से छोटी विषम प्राकृतिक संख्याओं का समुच्चय है, तो (A) और (B) का संबंध क्या है?
If \(A=\{1,3,5\}\) and (B) is the set of odd natural numbers less than (6), what is the relation between (A) and (B)?
#sets
#set-builder
#odd-numbers
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A समान / Equal
B अलग / Disjoint
C (A) बड़ा है / (A) is larger
D (B) रिक्त है / (B) is empty
Explanation opens after your attempt
Correct Answer
A. समान / Equal
Step 1
Concept
The odd natural numbers less than (6) are (1,3,5). Converting set-builder form to roster form is safest in exams.
Step 2
Why this answer is correct
The correct answer is A. समान / Equal. The odd natural numbers less than (6) are (1,3,5). Converting set-builder form to roster form is safest in exams.
Step 3
Exam Tip
(6) से छोटी विषम प्राकृतिक संख्याएँ (1,3,5) हैं। परिभाषा को सूची में बदलना परीक्षा में सबसे सुरक्षित तरीका है।
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यदि \(A=\{a,b\}\) और \(B=\{a,b,c\}\) हैं, तो कौन सा कथन सत्य है?
If \(A=\{a,b\}\) and \(B=\{a,b,c\}\), which statement is true?
#subsets
#proper-subset
#letters
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A \(A\subset B\)
B \(B\subset A\)
C (A=B)
D \(A\not\subseteq B\)
Explanation opens after your attempt
Correct Answer
A. \(A\subset B\)
Step 1
Concept
Every member of (A) is in (B), and (B) has one extra member. Hence (A) is a proper subset.
Step 2
Why this answer is correct
The correct answer is A. \(A\subset B\). Every member of (A) is in (B), and (B) has one extra member. Hence (A) is a proper subset.
Step 3
Exam Tip
(A) का हर सदस्य (B) में है और (B) में एक अतिरिक्त सदस्य है। इसलिए (A) उचित उपसमुच्चय है।
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तीन सदस्यों वाले समुच्चय के कुल उपसमुच्चयों की संख्या कितनी होती है?
How many subsets does a set with three elements have?
#subsets
#counting
#power-set
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets. Here \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). A set with (n) elements has \(2^n\) subsets. Here \(2^3=8\).
Step 3
Exam Tip
(n) सदस्यों वाले सेट के उपसमुच्चय \(2^n\) होते हैं। यहाँ \(2^3=8\) है।
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चार सदस्यों वाले समुच्चय के उचित उपसमुच्चयों की संख्या कितनी है?
How many proper subsets does a set with four elements have?
#subsets
#proper-subset
#counting
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A (15)
B (16)
C (8)
D (14)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\), and removing the whole set leaves (15) proper subsets. Do not forget to subtract the set itself.
Step 2
Why this answer is correct
The correct answer is A. (15). Total subsets are \(2^4=16\), and removing the whole set leaves (15) proper subsets. Do not forget to subtract the set itself.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं और पूरा सेट हटाने पर (15) उचित उपसमुच्चय बचते हैं। अंतिम उत्तर में पूरा सेट घटाना न भूलें।
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किस शर्त पर दो समुच्चय (A) और (B) बराबर कहलाते हैं?
Under which condition are two sets (A) and (B) called equal?
#equal-sets
#mutual-subset
#theory
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A \(A\subseteq B\) और \(B\subseteq A\) / \(A\subseteq B\) and \(B\subseteq A\)
B केवल \(A\subseteq B\) / Only \(A\subseteq B\)
C केवल (n(A)=n(B)) / Only (n(A)=n(B))
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq B\) और \(B\subseteq A\) / \(A\subseteq B\) and \(B\subseteq A\)
Step 1
Concept
If each set is a subset of the other, every member matches. Equal cardinality alone is not enough.
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\) और \(B\subseteq A\) / \(A\subseteq B\) and \(B\subseteq A\). If each set is a subset of the other, every member matches. Equal cardinality alone is not enough.
Step 3
Exam Tip
दोनों दिशाओं में उपसमुच्चय संबंध होने पर हर सदस्य समान होता है। केवल सदस्यों की संख्या बराबर होना काफी नहीं है।
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रिक्त समुच्चय के बारे में कौन सा कथन हमेशा सत्य है?
Which statement about the empty set is always true?
#empty-set
#subsets
#concept
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A \(\emptyset\subseteq A\)
B \(A\subseteq\emptyset\)
C \(\emptyset=A\)
D \(\emptyset\in A\)
Explanation opens after your attempt
Correct Answer
A. \(\emptyset\subseteq A\)
Step 1
Concept
The empty set is a subset of every set. It has no element, so no counterexample exists.
Step 2
Why this answer is correct
The correct answer is A. \(\emptyset\subseteq A\). The empty set is a subset of every set. It has no element, so no counterexample exists.
Step 3
Exam Tip
रिक्त समुच्चय हर समुच्चय का उपसमुच्चय होता है। इसमें कोई सदस्य नहीं होता, इसलिए विरोधी उदाहरण नहीं मिल सकता।
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यदि \(A=\emptyset\) है, तो (A) के उपसमुच्चयों की संख्या क्या होगी?
If \(A=\emptyset\), how many subsets does (A) have?
#empty-set
#counting
#subsets
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A (0)
B (1)
C (2)
D अनंत / Infinite
Explanation opens after your attempt
Step 1
Concept
The empty set has only one subset, itself. The formula gives \(2^0=1\).
Step 2
Why this answer is correct
The correct answer is B. (1). The empty set has only one subset, itself. The formula gives \(2^0=1\).
Step 3
Exam Tip
रिक्त समुच्चय का केवल एक उपसमुच्चय वही स्वयं है। सूत्र \(2^0=1\) देता है।
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यदि \(A={\emptyset}\) है, तो कौन सा कथन सत्य है?
If \(A={\emptyset}\), which statement is true?
#empty-set
#membership
#subsets
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A \(\emptyset\in A\)
B \(\emptyset=A\)
C \(A=\emptyset\)
D (A) का कोई सदस्य नहीं / (A) has no element
Explanation opens after your attempt
Correct Answer
A. \(\emptyset\in A\)
Step 1
Concept
Here (A) has one element, namely \(\emptyset\). Keep \(\emptyset\) and \({\emptyset}\) distinct.
Step 2
Why this answer is correct
The correct answer is A. \(\emptyset\in A\). Here (A) has one element, namely \(\emptyset\). Keep \(\emptyset\) and \({\emptyset}\) distinct.
Step 3
Exam Tip
यहाँ (A) में एक सदस्य है और वह \(\emptyset\) है। \(\emptyset\) और \({\emptyset}\) को अलग रखें।
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यदि \(A=\{1,2,{1,2}\}\) है, तो कौन सा कथन सत्य है?
If \(A=\{1,2,{1,2}\}\), which statement is true?
#membership
#subsets
#nested-set
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A \({1,2}\in A\)
B ({1,2}=A)
C \(3\in A\)
D \({1,2,3}\subseteq A\)
Explanation opens after your attempt
Correct Answer
A. \({1,2}\in A\)
Step 1
Concept
Here ({1,2}) is listed as one element. Distinguish membership from subset relation.
Step 2
Why this answer is correct
The correct answer is A. \({1,2}\in A\). Here ({1,2}) is listed as one element. Distinguish membership from subset relation.
Step 3
Exam Tip
({1,2}) यहाँ एक सदस्य के रूप में दिया है। सदस्यता और उपसमुच्चय को अलग पहचानें।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो कितने समुच्चय (X) ऐसे हैं कि \(A\subseteq X\subseteq B\)?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many sets (X) satisfy \(A\subseteq X\subseteq B\)?
#subsets
#between-sets
#counting
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The elements of (A) are fixed in (X), while (4,5) are optional. Hence \(2^2=4\) sets are possible.
Step 2
Why this answer is correct
The correct answer is B. (4). The elements of (A) are fixed in (X), while (4,5) are optional. Hence \(2^2=4\) sets are possible.
Step 3
Exam Tip
(X) में (A) के सदस्य निश्चित हैं और (4,5) वैकल्पिक हैं। इसलिए \(2^2=4\) समुच्चय बनेंगे।
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यदि \(A=\{1,2,3,4\}\) है, तो (2) को शामिल करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets contain (2)?
#subsets
#counting
#fixed-element
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Element (2) is fixed, and the remaining (3) elements are optional. Therefore the count is \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). Element (2) is fixed, and the remaining (3) elements are optional. Therefore the count is \(2^3=8\).
Step 3
Exam Tip
(2) निश्चित है और बाकी (3) सदस्य वैकल्पिक हैं। इसलिए संख्या \(2^3=8\) है।
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यदि \(A=\{p,q,r,s,t\}\) है, तो (p) को न शामिल करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{p,q,r,s,t\}\), how many subsets do not contain (p)?
#subsets
#counting
#excluded-element
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A (8)
B (16)
C (31)
D (32)
Explanation opens after your attempt
Step 1
Concept
After excluding (p), (4) elements remain free. Thus \(2^4=16\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (16). After excluding (p), (4) elements remain free. Thus \(2^4=16\) subsets are possible.
Step 3
Exam Tip
(p) हटाने के बाद (4) सदस्य स्वतंत्र रहते हैं। इसलिए \(2^4=16\) उपसमुच्चय बनते हैं।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (1,2) को शामिल और (5) को बाहर रखने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets contain (1,2) and exclude (5)?
#subsets
#counting
#conditions
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Elements (1,2) are fixed, (5) is forbidden, and (3,4) are optional. Hence the answer is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). Elements (1,2) are fixed, (5) is forbidden, and (3,4) are optional. Hence the answer is \(2^2=4\).
Step 3
Exam Tip
(1,2) निश्चित हैं, (5) निषिद्ध है और (3,4) वैकल्पिक हैं। इसलिए \(2^2=4\) उत्तर है।
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यदि \(A={x:x^2-5x+6=0}\) और \(B=\{2,3\}\) हैं, तो कौन सा संबंध सही है?
If \(A={x:x^2-5x+6=0}\) and \(B=\{2,3\}\), which relation is correct?
#equal-sets
#solution-set
#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 of the equation are (2) and (3). The solution set is the same as (B).
Step 2
Why this answer is correct
The correct answer is A. (A=B). The roots of the equation are (2) and (3). The solution set is the same as (B).
Step 3
Exam Tip
समीकरण के मूल (2) और (3) हैं। समाधान समुच्चय (B) के समान है।
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यदि \(A={x:x^2=9,\ x\in\mathbb{Z}}\) और \(B=\{-3,3\}\) हैं, तो क्या सत्य है?
If \(A={x:x^2=9,\ x\in\mathbb{Z}}\) and \(B=\{-3,3\}\), what is true?
#equal-sets
#integers
#solution-set
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A (A=B)
B \(A=\{3\}\)
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 (-3) and (3). Missing the negative solution is a common error.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The integer solutions are both (-3) and (3). Missing the negative solution is a common error.
Step 3
Exam Tip
पूर्णांक हल (-3) और (3) दोनों हैं। ऋणात्मक हल को छोड़ना सामान्य गलती है।
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यदि \(A=\{1,2,3\}\) है और (B), (A) के उन सदस्यों का समुच्चय है जो (3) से छोटे या बराबर हैं, तो सही विकल्प चुनिए।
If \(A=\{1,2,3\}\) and (B) is the set of elements of (A) that are less than or equal to (3), choose the correct option.
#equal-sets
#subset-condition
#reasoning
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A (A=B)
B \(B=\emptyset\)
C \(A\subset B\)
D \(B\subset A\) उचित / \(B\subset A\) proper
Explanation opens after your attempt
Step 1
Concept
Every element of (A) is less than or equal to (3). Therefore (B) has the same elements.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Every element of (A) is less than or equal to (3). Therefore (B) has the same elements.
Step 3
Exam Tip
(A) का हर सदस्य (3) से कम या बराबर है। इसलिए (B) में भी वही सदस्य हैं।
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यदि \(A\subseteq B\) और \(B\subseteq C\) हैं, तो कौन सा निष्कर्ष हमेशा सही है?
If \(A\subseteq B\) and \(B\subseteq C\), which conclusion is always true?
#subsets
#transitivity
#logic
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A \(A\subseteq C\)
B \(C\subseteq A\)
C (A=C)
D \(B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq C\)
Step 1
Concept
The subset relation is transitive. Elements pass from (A) through (B) into (C).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq C\). The subset relation is transitive. Elements pass from (A) through (B) into (C).
Step 3
Exam Tip
उपसमुच्चय संबंध संक्रमणीय होता है। बीच वाले समुच्चय से होकर सदस्य (C) तक पहुंचते हैं।
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यदि \(A\subset B\) और \(B\subset C\) हैं, तो (A) और (C) के बारे में कौन सा कथन सही है?
If \(A\subset B\) and \(B\subset C\), which statement about (A) and (C) is correct?
#proper-subset
#transitivity
#logic
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A \(A\subset C\)
B (A=C)
C \(C\subset A\)
D \(A\not\subseteq C\)
Explanation opens after your attempt
Correct Answer
A. \(A\subset C\)
Step 1
Concept
With two proper inclusions, (C) must have more than (A). Hence (A) is a proper subset of (C).
Step 2
Why this answer is correct
The correct answer is A. \(A\subset C\). With two proper inclusions, (C) must have more than (A). Hence (A) is a proper subset of (C).
Step 3
Exam Tip
दो उचित बढ़ाव होने से (C) में (A) से अधिक सदस्य जरूर होंगे। इसलिए (A) (C) का उचित उपसमुच्चय है।
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यदि \(A\subseteq B\) और (n(A)=n(B)) है, जहाँ दोनों सीमित समुच्चय हैं, तो क्या निष्कर्ष है?
If \(A\subseteq B\) and (n(A)=n(B)), where both sets are finite, what follows?
#equal-sets
#finite-sets
#cardinality
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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 कोई निष्कर्ष नहीं / No conclusion
Explanation opens after your attempt
Step 1
Concept
For finite sets, if a subset has the same number of elements, no extra element remains. Therefore the sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). For finite sets, if a subset has the same number of elements, no extra element remains. Therefore the sets are equal.
Step 3
Exam Tip
सीमित समुच्चयों में उपसमुच्चय की सदस्य संख्या बराबर हो तो कोई अतिरिक्त सदस्य नहीं बचता। इसलिए समुच्चय बराबर हैं।
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यदि (A) और (B) सीमित हैं और (n(A)=n(B)) है, तो क्या (A=B) हमेशा होगा?
If (A) and (B) are finite and (n(A)=n(B)), must (A=B) always hold?
#equal-sets
#cardinality
#common-mistake
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A नहीं / No
B हाँ / Yes
C केवल जब दोनों रिक्त हों / Only if both are empty
D केवल जब दोनों अनंत हों / Only if both are infinite
Explanation opens after your attempt
Correct Answer
A. नहीं / No
Step 1
Concept
Equal number of elements does not mean the elements are the same. For example, ({1,2}) and ({3,4}) are not equal.
Step 2
Why this answer is correct
The correct answer is A. नहीं / No. Equal number of elements does not mean the elements are the same. For example, ({1,2}) and ({3,4}) are not equal.
Step 3
Exam Tip
सदस्यों की संख्या बराबर होने से सदस्य समान होना जरूरी नहीं है। जैसे ({1,2}) और ({3,4}) बराबर नहीं हैं।
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यदि \(A=\{1,2\}\) है, तो (\mathcal{P}(A)) कौन सा है?
If \(A=\{1,2\}\), which is (\mathcal{P}(A))?
#power-set
#subsets
#roster
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A \({\emptyset,{1},{2},{1,2}}\)
B \({1,2,\emptyset}\)
C ({{1},{2}})
D ({1,2})
Explanation opens after your attempt
Correct Answer
A. \({\emptyset,{1},{2},{1,2}}\)
Step 1
Concept
The power set contains all subsets as elements. Include both the empty set and the whole set.
Step 2
Why this answer is correct
The correct answer is A. \({\emptyset,{1},{2},{1,2}}\). The power set contains all subsets as elements. Include both the empty set and the whole set.
Step 3
Exam Tip
घात समुच्चय में सभी उपसमुच्चय सदस्य बनते हैं। रिक्त समुच्चय और पूरा समुच्चय दोनों शामिल करें।
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यदि \(A=\{1,2,3\}\) है, तो कौन सा (\mathcal{P}(A)) का सदस्य है?
If \(A=\{1,2,3\}\), which is an element of (\mathcal{P}(A))?
#power-set
#membership
#subsets
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A ({1,3})
B (1)
C (4)
D ({{4}})
Explanation opens after your attempt
Correct Answer
A. ({1,3})
Step 1
Concept
Elements of (\mathcal{P}(A)) are subsets of (A). Since \({1,3}\subseteq A\), it is an element.
Step 2
Why this answer is correct
The correct answer is A. ({1,3}). Elements of (\mathcal{P}(A)) are subsets of (A). Since \({1,3}\subseteq A\), it is an element.
Step 3
Exam Tip
(\mathcal{P}(A)) के सदस्य (A) के उपसमुच्चय होते हैं। \({1,3}\subseteq A\) है, इसलिए यह सदस्य है।
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यदि (A=B) है, तो (\mathcal{P}(A)) और (\mathcal{P}(B)) के बारे में क्या सत्य है?
If (A=B), what is true about (\mathcal{P}(A)) and (\mathcal{P}(B))?
#equal-sets
#power-set
#theory
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A (\mathcal{P}(A)=\mathcal{P}(B))
B (\mathcal{P}(A)\subset \mathcal{P}(B)) उचित / (\mathcal{P}(A)\subset \mathcal{P}(B)) proper
C वे असंबंधित हैं / They are unrelated
D दोनों रिक्त हैं / Both are empty
Explanation opens after your attempt
Correct Answer
A. (\mathcal{P}(A)=\mathcal{P}(B))
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}(B)). Equal sets have exactly the same subsets. Therefore their power sets are equal.
Step 3
Exam Tip
बराबर समुच्चयों के सभी उपसमुच्चय भी समान होते हैं। इसलिए उनके घात समुच्चय बराबर होते हैं।
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यदि (\mathcal{P}(A)=\mathcal{P}(B)) है, तो कौन सा निष्कर्ष सही है?
If (\mathcal{P}(A)=\mathcal{P}(B)), which conclusion is correct?
#power-set
#equal-sets
#reasoning
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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
A set itself appears in its power set. Equal power sets force the original sets to be equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). A set itself appears in its power set. Equal power sets force the original sets to be equal.
Step 3
Exam Tip
किसी समुच्चय का पूरा समुच्चय स्वयं उसके घात समुच्चय में आता है। बराबर घात समुच्चय मूल समुच्चय भी बराबर बनाते हैं।
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यदि \(A\subseteq B\), तो कौन सा कथन हमेशा सत्य है?
If \(A\subseteq B\), which statement is always true?
#subsets
#union
#identity
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A \(A\cup B=B\)
B \(A\cup B=A\)
C \(A\cap B=B\)
D (A-B=B)
Explanation opens after your attempt
Correct Answer
A. \(A\cup B=B\)
Step 1
Concept
When (A) lies inside (B), union adds no new element to (B). Hence \(A\cup B=B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\cup B=B\). When (A) lies inside (B), union adds no new element to (B). Hence \(A\cup B=B\).
Step 3
Exam Tip
जब (A) पूरी तरह (B) के अंदर है, तो मिलन में कोई नया सदस्य नहीं जुड़ता। इसलिए \(A\cup B=B\) है।
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यदि \(A\subseteq B\) है, तो प्रतिच्छेद के लिए कौन सा कथन हमेशा सत्य है?
If \(A\subseteq B\), which statement about intersection is always true?
#subsets
#intersection
#identity
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A \(A\cap B=A\)
B \(A\cap B=B\)
C (A-B=A)
D \(B-A=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(A\cap B=A\)
Step 1
Concept
Every element of (A) is in (B), so the common part is (A). A quick Venn diagram helps identify this.
Step 2
Why this answer is correct
The correct answer is A. \(A\cap B=A\). Every element of (A) is in (B), so the common part is (A). A quick Venn diagram helps identify this.
Step 3
Exam Tip
(A) का हर सदस्य (B) में है, इसलिए साझा भाग (A) ही है। चित्र बनाकर यह पहचान जल्दी होती है।
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यदि \(A\cup B=B\), तो कौन सा निष्कर्ष हमेशा सही है?
If \(A\cup B=B\), which conclusion is always correct?
#subsets
#union
#reverse-reasoning
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A \(A\subseteq B\)
B \(B\subseteq A\)
C \(A=B^c\)
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq B\)
Step 1
Concept
The union remains (B), so (A) added no outside element. Therefore \(A\subseteq B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\). The union remains (B), so (A) added no outside element. Therefore \(A\subseteq B\).
Step 3
Exam Tip
मिलन (B) ही रह गया, इसका अर्थ है (A) ने बाहर का कोई सदस्य नहीं जोड़ा। इसलिए \(A\subseteq B\) है।
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यदि \(A\cap B=A\), तो कौन सा निष्कर्ष हमेशा सही है?
If \(A\cap B=A\), which conclusion is always correct?
#subsets
#intersection
#reverse-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 common part is (A), so every element of (A) is in (B). This is exactly subset relation.
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\). The common part is (A), so every element of (A) is in (B). This is exactly subset relation.
Step 3
Exam Tip
साझा भाग (A) है, इसलिए (A) का हर सदस्य (B) में है। यह उपसमुच्चय की पहचान है।
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यदि \(A-B=\emptyset\), तो कौन सा निष्कर्ष सही है?
If \(A-B=\emptyset\), which conclusion is correct?
#subsets
#difference
#logic
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A \(A\subseteq B\)
B \(B\subseteq A\)
C (A=B) हमेशा / (A=B) always
D \(A\cap B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq B\)
Step 1
Concept
There is no element in (A) outside (B). Hence \(A\subseteq B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\). There is no element in (A) outside (B). Hence \(A\subseteq B\).
Step 3
Exam Tip
(A) में ऐसा कोई सदस्य नहीं जो (B) से बाहर हो। इसलिए \(A\subseteq B\) है।
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यदि \(A\triangle B=\emptyset\), तो कौन सा निष्कर्ष सही है?
If \(A\triangle B=\emptyset\), which conclusion is correct?
#equal-sets
#symmetric-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 दोनों असंबद्ध / Both disjoint
Explanation opens after your attempt
Step 1
Concept
If the symmetric difference is empty, no element belongs to exactly one set. Thus the sets have identical elements.
Step 2
Why this answer is correct
The correct answer is A. (A=B). If the symmetric difference is empty, no element belongs to exactly one set. Thus the sets have identical elements.
Step 3
Exam Tip
सममित अंतर रिक्त है तो कोई सदस्य केवल एक ही सेट में नहीं है। इसलिए दोनों सेटों के सदस्य समान हैं।
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यदि (A) संख्या (18) के अभाज्य भाजकों का समुच्चय है और \(B=\{2,3\}\), तो सही कथन क्या है?
If (A) is the set of prime divisors of (18) and \(B=\{2,3\}\), what is correct?
#equal-sets
#prime-divisors
#set-builder
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A (A=B)
B \(A=\{2,3,9\}\)
C \(B\subset A\) उचित / \(B\subset A\) proper
D \(A=\emptyset\)
Explanation opens after your attempt
Step 1
Concept
The prime divisors of (18) are only (2) and (3). Note the difference between divisors and prime divisors.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The prime divisors of (18) are only (2) and (3). Note the difference between divisors and prime divisors.
Step 3
Exam Tip
(18) के अभाज्य भाजक केवल (2) और (3) हैं। भाजक और अभाज्य भाजक में अंतर ध्यान रखें।
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यदि (A) संख्या (24) के सम भाजकों का समुच्चय है, तो कौन सा (A) का उपसमुच्चय है?
If (A) is the set of even divisors of (24), which is a subset of (A)?
#subsets
#divisors
#application
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A ({2,4,6,8})
B ({2,4,5})
C ({1,2,4})
D ({10,12})
Explanation opens after your attempt
Correct Answer
A. ({2,4,6,8})
Step 1
Concept
Here \(A=\{2,4,6,8,12,24\}\). Write the full set first, then test every option.
Step 2
Why this answer is correct
The correct answer is A. ({2,4,6,8}). Here \(A=\{2,4,6,8,12,24\}\). Write the full set first, then test every option.
Step 3
Exam Tip
\(A=\{2,4,6,8,12,24\}\) है। पहले पूरे सेट को लिखें, फिर हर विकल्प जांचें।
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यदि \(A=\{2,3,5,7\}\) है, तो कितने उपसमुच्चय (2) और (7) दोनों को शामिल करते हैं?
If \(A=\{2,3,5,7\}\), how many subsets contain both (2) and (7)?
#subsets
#counting
#fixed-elements
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Elements (2,7) are fixed, while (3,5) are optional. So there are \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. (4). Elements (2,7) are fixed, while (3,5) are optional. So there are \(2^2=4\) subsets.
Step 3
Exam Tip
(2,7) निश्चित हैं और (3,5) वैकल्पिक हैं। इसलिए \(2^2=4\) उपसमुच्चय हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो केवल विषम सदस्यों से बने उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets can be formed using only odd elements?
#subsets
#counting
#odd-elements
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The odd elements are (1,3,5). Their subsets are \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). The odd elements are (1,3,5). Their subsets are \(2^3=8\).
Step 3
Exam Tip
विषम सदस्य (1,3,5) हैं। इनके उपसमुच्चय \(2^3=8\) होंगे।
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यदि \(A=\{1,2,3,4\}\) है, तो कम से कम एक सदस्य वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets have at least one element?
#subsets
#counting
#non-empty
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A (14)
B (15)
C (16)
D (8)
Explanation opens after your attempt
Step 1
Concept
Total subsets are (16), and only the empty set is excluded. Hence (16-1=15).
Step 2
Why this answer is correct
The correct answer is B. (15). Total subsets are (16), and only the empty set is excluded. Hence (16-1=15).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और केवल रिक्त समुच्चय हटाना है। इसलिए (16-1=15) है।
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यदि (A) में (3) सदस्य हैं, तो (\mathcal{P}(A)) के उचित उपसमुच्चयों की संख्या कितनी है?
If (A) has (3) elements, how many proper subsets does (\mathcal{P}(A)) have?
#power-set
#counting
#proper-subset
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A (127)
B (255)
C (256)
D (511)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^3=8\) elements. Its proper subsets are \(2^8-1=255\).
Step 2
Why this answer is correct
The correct answer is B. (255). (\mathcal{P}(A)) has \(2^3=8\) elements. Its proper subsets are \(2^8-1=255\).
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^3=8\) सदस्य हैं। इसके उचित उपसमुच्चय \(2^8-1=255\) होंगे।
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यदि (A) में (3) सदस्य हैं, तो कितने क्रमित युग्म ((X,Y)) संभव हैं जिनमें \(X\subseteq Y\subseteq A\)?
If (A) has (3) elements, how many ordered pairs ((X,Y)) are possible such that \(X\subseteq Y\subseteq A\)?
#subsets
#counting
#ordered-pairs
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A (9)
B (18)
C (27)
D (64)
Explanation opens after your attempt
Step 1
Concept
Each element has three choices: in neither set, in (Y) only, or in both. Thus the total is \(3^3=27\).
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. Thus the total is \(3^3=27\).
Step 3
Exam Tip
हर सदस्य के लिए तीन स्थितियां हैं: दोनों में नहीं, केवल (Y) में, या दोनों में। इसलिए कुल \(3^3=27\) है।
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यदि \(A\subset B\) और (B-A={9}) है, तो कितने समुच्चय (X) ऐसे हैं कि \(A\subseteq X\subseteq B\)?
If \(A\subset B\) and (B-A={9}), how many sets (X) satisfy \(A\subseteq X\subseteq B\)?
#subsets
#between-sets
#difference
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
Only (9) is optional outside (A). Therefore (X) can be (A) or (B).
Step 2
Why this answer is correct
The correct answer is B. (2). Only (9) is optional outside (A). Therefore (X) can be (A) or (B).
Step 3
Exam Tip
(A) के बाहर केवल (9) वैकल्पिक है। इसलिए (X=A) या (X=B) हो सकता है।
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यदि ({2a+1}={7}), तो (a) का मान क्या है?
If ({2a+1}={7}), what is the value of (a)?
#equal-sets
#singleton
#parameter
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The singleton sets are equal, so (2a+1=7). This gives (a=3).
Step 2
Why this answer is correct
The correct answer is B. (3). The singleton sets are equal, so (2a+1=7). This gives (a=3).
Step 3
Exam Tip
एक-सदस्यीय समुच्चय बराबर हैं, इसलिए (2a+1=7)। इससे (a=3) मिलता है।
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यदि ({a,4}={2,b}) और \(a\ne4\), तो सही युग्म कौन सा है?
If ({a,4}={2,b}) and \(a\ne4\), which pair is correct?
#equal-sets
#parameter
#two-element-set
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A (a=2,\ b=4)
B (a=4,\ b=2)
C (a=2,\ b=2)
D (a=4,\ b=4)
Explanation opens after your attempt
Correct Answer
A. (a=2,\ b=4)
Step 1
Concept
Equal sets must contain both (2) and (4). Since \(a\ne4\), (a=2) and then (b=4).
Step 2
Why this answer is correct
The correct answer is A. (a=2,\ b=4). Equal sets must contain both (2) and (4). Since \(a\ne4\), (a=2) and then (b=4).
Step 3
Exam Tip
बराबर समुच्चयों में (2) और (4) दोनों सदस्य होने चाहिए। \(a\ne4\) से (a=2) और फिर (b=4) होगा।
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यदि \({1,k}\subseteq{1,2,3}\), तो (k) के संभावित मानों का समुच्चय क्या है?
If \({1,k}\subseteq{1,2,3}\), what is the set of possible values of (k)?
#subsets
#parameter
#reasoning
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A ({2,3})
B ({1,2,3})
C ({0,1,2,3})
D ({1})
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Correct Answer
B. ({1,2,3})
Step 1
Concept
The value (k) must be an element of the larger set. If (k=1), the set becomes ({1}), which is still a subset.
Step 2
Why this answer is correct
The correct answer is B. ({1,2,3}). The value (k) must be an element of the larger set. If (k=1), the set becomes ({1}), which is still a subset.
Step 3
Exam Tip
(k) भी बड़े समुच्चय का सदस्य होना चाहिए। (k=1) होने पर समुच्चय ({1}) बनता है, जो उपसमुच्चय है।
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यदि \(A=\{0,1,2,3\}\), तो (0) को शामिल करने वाले ठीक दो-सदस्यीय उपसमुच्चयों की संख्या कितनी है?
If \(A=\{0,1,2,3\}\), how many two-element subsets contain (0)?
#subsets
#counting
#two-element
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
Element (0) is fixed, and the second element is chosen from (1,2,3). So there are (3) subsets.
Step 2
Why this answer is correct
The correct answer is B. (3). Element (0) is fixed, and the second element is chosen from (1,2,3). So there are (3) subsets.
Step 3
Exam Tip
(0) निश्चित है और दूसरा सदस्य (1,2,3) में से चुना जाएगा। इसलिए (3) उपसमुच्चय बनते हैं।
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यदि \(A={\emptyset,{\emptyset}}\) है, तो (A) के उपसमुच्चयों की संख्या कितनी है?
If \(A={\emptyset,{\emptyset}}\), how many subsets does (A) have?
#empty-set
#nested-set
#counting
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A (2)
B (3)
C (4)
D (5)
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Step 1
Concept
The set (A) has two distinct elements: \(\emptyset\) and \({\emptyset}\). Hence it has \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is C. (4). The set (A) has two distinct elements: \(\emptyset\) and \({\emptyset}\). Hence it has \(2^2=4\) subsets.
Step 3
Exam Tip
(A) में दो अलग सदस्य हैं: \(\emptyset\) और \({\emptyset}\)। इसलिए उपसमुच्चय \(2^2=4\) हैं।
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यदि \(A={\emptyset,{\emptyset},1}\) है, तो कौन सा (A) का उपसमुच्चय नहीं है?
If \(A={\emptyset,{\emptyset},1}\), which is not a subset of (A)?
#subsets
#nested-set
#not-subset
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A \({\emptyset,1}\)
B \({{\emptyset}}\)
C \({\emptyset,{\emptyset},1}\)
D ({1,2})
Explanation opens after your attempt
Correct Answer
D. ({1,2})
Step 1
Concept
The element (2) is not in (A). Every element of a subset must belong to the original set.
Step 2
Why this answer is correct
The correct answer is D. ({1,2}). The element (2) is not in (A). Every element of a subset must belong to the original set.
Step 3
Exam Tip
(2) समुच्चय (A) का सदस्य नहीं है। उपसमुच्चय में हर सदस्य मूल समुच्चय में होना चाहिए।
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यदि \(A\subseteq B\) है, तो पूरकों के लिए कौन सा कथन सही है?
If \(A\subseteq B\), which statement about complements is correct?
#subsets
#complement
#logic
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A \(B^c\subseteq A^c\)
B \(A^c\subseteq B^c\)
C \(A^c=B^c\) हमेशा / \(A^c=B^c\) always
D \(A^c\cap B^c=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \(B^c\subseteq A^c\)
Step 1
Concept
The complement of the larger set is smaller. The subset relation reverses under complements.
Step 2
Why this answer is correct
The correct answer is A. \(B^c\subseteq A^c\). The complement of the larger set is smaller. The subset relation reverses under complements.
Step 3
Exam Tip
बड़े समुच्चय का पूरक छोटा होता है। उपसमुच्चय संबंध पूरक लेने पर उलट जाता है।
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कथन: यदि \(A\subset B\), तो (\mathcal{P}(A)\subset \mathcal{P}(B))। कारण: (A) का हर उपसमुच्चय (B) का भी उपसमुच्चय है और (B) स्वयं (\mathcal{P}(B)) में है पर (\mathcal{P}(A)) में नहीं। सही विकल्प चुनिए।
Assertion: If \(A\subset B\), then (\mathcal{P}(A)\subset \mathcal{P}(B)). Reason: Every subset of (A) is also a subset of (B), and (B) itself is in (\mathcal{P}(B)) but not in (\mathcal{P}(A)). Choose the correct option.
#assertion-reason
#power-set
#proper-subset
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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
A proper subset gives a proper inclusion of power sets. The reason correctly explains both inclusion and properness.
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. A proper subset gives a proper inclusion of power sets. The reason correctly explains both inclusion and properness.
Step 3
Exam Tip
उचित उपसमुच्चय होने पर घात समुच्चय भी उचित रूप से शामिल होता है। कारण में दोनों भाग सही व्याख्या देते हैं।
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