यदि \(A=\{2,4,6,8,10\}\) है, तो (\mathcal{P}(A)) में कुल कितने तत्व होंगे?
If \(A=\{2,4,6,8,10\}\), how many elements will (\mathcal{P}(A)) have?
#sets
#power set
#cardinality
#class 11
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A (16)
B (25)
C (32)
D (10)
Explanation opens after your attempt
Step 1
Concept
Set (A) has (5) elements, so its power set has \(2^5=32\) elements. In exams, first count the elements of the original set.
Step 2
Why this answer is correct
The correct answer is C. (32). Set (A) has (5) elements, so its power set has \(2^5=32\) elements. In exams, first count the elements of the original set.
Step 3
Exam Tip
(A) में (5) तत्व हैं, इसलिए घात समुच्चय में \(2^5=32\) तत्व होंगे। परीक्षा में पहले मूल समुच्चय के तत्व गिनें।
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यदि \(A=\{1,3,5\}\) है, तो (\mathcal{P}(A)) में ऐसे कितने उपसमुच्चय होंगे जिनमें (3) अवश्य हो?
If \(A=\{1,3,5\}\), how many subsets in (\mathcal{P}(A)) must contain (3)?
#sets
#power set
#subset counting
#class 11
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
Keeping (3) is fixed and the remaining (1,5) may be chosen or not chosen. So the number is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). Keeping (3) is fixed and the remaining (1,5) may be chosen or not chosen. So the number is \(2^2=4\).
Step 3
Exam Tip
(3) को रखना निश्चित है और बाकी (1,5) चुने या छोड़े जा सकते हैं। इसलिए संख्या \(2^2=4\) होगी।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\) और \(A=\{2,3,5,7\}\) है, तो (A') क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{2,3,5,7\}\), what is (A')?
#sets
#universal set
#complement
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A ({1,4,6,8})
B ({2,3,5,7})
C ({1,2,4,6})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({1,4,6,8})
Step 1
Concept
(A') contains the elements of (U) that are not in (A). Therefore (1,4,6,8) form the correct complement.
Step 2
Why this answer is correct
The correct answer is A. ({1,4,6,8}). (A') contains the elements of (U) that are not in (A). Therefore (1,4,6,8) form the correct complement.
Step 3
Exam Tip
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। इसलिए (1,4,6,8) सही पूरक हैं।
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यदि (n(\mathcal{P}(B))=128) है, तो (B) के उचित उपसमुच्चयों की संख्या कितनी होगी?
If (n(\mathcal{P}(B))=128), how many proper subsets does (B) have?
#sets
#power set
#proper subsets
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A (7)
B (64)
C (127)
D (128)
Explanation opens after your attempt
Step 1
Concept
There are (128) total subsets, and a proper subset does not include the whole set itself. So the number is (128-1=127).
Step 2
Why this answer is correct
The correct answer is C. (127). There are (128) total subsets, and a proper subset does not include the whole set itself. So the number is (128-1=127).
Step 3
Exam Tip
कुल उपसमुच्चय (128) हैं और उचित उपसमुच्चय में पूरा समुच्चय नहीं गिना जाता। इसलिए संख्या (128-1=127) होगी।
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यदि \(A=\{a,{b},c\}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व नहीं है?
If \(A=\{a,{b},c\}\), which of the following is not an element of (\mathcal{P}(A))?
#sets
#power set
#element subset
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A ({a,c})
B ({{b}})
C ({a,{b}})
D ({a,b})
Explanation opens after your attempt
Correct Answer
D. ({a,b})
Step 1
Concept
(b) itself is not an element of (A); ({b}) is an element. So ({a,b}) is not a subset of (A).
Step 2
Why this answer is correct
The correct answer is D. ({a,b}). (b) itself is not an element of (A); ({b}) is an element. So ({a,b}) is not a subset of (A).
Step 3
Exam Tip
(b) स्वयं (A) का तत्व नहीं है, बल्कि ({b}) तत्व है। इसलिए ({a,b}), (A) का उपसमुच्चय नहीं है।
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यदि \(A=\{p,q,r\}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व है?
If \(A=\{p,q,r\}\), which of the following is an element of (\mathcal{P}(A))?
#sets
#power set
#subset
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A (p)
B ({p,q})
C ({p,s})
D ({q,r,s})
Explanation opens after your attempt
Correct Answer
B. ({p,q})
Step 1
Concept
An element of a power set is always a subset of the original set. ({p,q}) contains only elements of (A), so it is correct.
Step 2
Why this answer is correct
The correct answer is B. ({p,q}). An element of a power set is always a subset of the original set. ({p,q}) contains only elements of (A), so it is correct.
Step 3
Exam Tip
घात समुच्चय का तत्व हमेशा मूल समुच्चय का उपसमुच्चय होता है। ({p,q}) में केवल (A) के तत्व हैं, इसलिए यह सही है।
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यदि \(U={x:x\in \mathbb{N}, x\leq 15}\) और \(A={x:x\) (5) का गुणज है(}), तो (n(A')) कितना होगा?
If \(U={x:x\in \mathbb{N}, x\leq 15}\) and \(A={x:x\) is a multiple of (5)(}), what is (n(A'))?
#sets
#universal set
#cardinality
#class 11
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A (3)
B (5)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(A=\{5,10,15\}\), so (n(A)=3). Since (U) has (15) elements, (n(A')=15-3=12).
Step 2
Why this answer is correct
The correct answer is D. (12). \(A=\{5,10,15\}\), so (n(A)=3). Since (U) has (15) elements, (n(A')=15-3=12).
Step 3
Exam Tip
\(A=\{5,10,15\}\) है, इसलिए (n(A)=3)। (U) में (15) तत्व हैं, अतः (n(A')=15-3=12)।
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यदि \(A=\{1,2,{3}\}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व नहीं है?
If \(A=\{1,2,{3}\}\), which of the following is not an element of (\mathcal{P}(A))?
#sets
#power set
#common mistake
#class 11
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A ({1,2})
B ({{3}})
C ({1,{3}})
D ({1,3})
Explanation opens after your attempt
Correct Answer
D. ({1,3})
Step 1
Concept
(3) itself is not an element of (A); ({3}) is an element. So ({1,3}) is not a subset of (A).
Step 2
Why this answer is correct
The correct answer is D. ({1,3}). (3) itself is not an element of (A); ({3}) is an element. So ({1,3}) is not a subset of (A).
Step 3
Exam Tip
(3) स्वयं (A) का तत्व नहीं है, बल्कि ({3}) तत्व है। इसलिए ({1,3}) (A) का उपसमुच्चय नहीं है।
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रिक्त समुच्चय \(\varnothing\) के घात समुच्चय में कितने तत्व होते हैं?
How many elements are there in the power set of the empty set \(\varnothing\)?
#sets
#empty set
#power set
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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 (0) elements, so (\mathcal{P}\(\varnothing\)) has \(2^0=1\) element. Remember that \(\varnothing\) is also a subset.
Step 2
Why this answer is correct
The correct answer is B. (1). The empty set has (0) elements, so (\mathcal{P}\(\varnothing\)) has \(2^0=1\) element. Remember that \(\varnothing\) is also a subset.
Step 3
Exam Tip
रिक्त समुच्चय में (0) तत्व हैं, इसलिए (\mathcal{P}\(\varnothing\)) में \(2^0=1\) तत्व होता है। याद रखें कि \(\varnothing\) भी एक उपसमुच्चय है।
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(\mathcal{P}(\mathcal{P}\(\varnothing\))) में कितने तत्व होंगे?
How many elements will (\mathcal{P}(\mathcal{P}\(\varnothing\))) have?
#sets
#iterated power set
#empty set
#class 11
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}\(\varnothing\)={\varnothing}) has (1) element. Therefore its power set has \(2^1=2\) elements.
Step 2
Why this answer is correct
The correct answer is B. (2). (\mathcal{P}\(\varnothing\)={\varnothing}) has (1) element. Therefore its power set has \(2^1=2\) elements.
Step 3
Exam Tip
(\mathcal{P}\(\varnothing\)={\varnothing}) में (1) तत्व है। इसलिए उसके घात समुच्चय में \(2^1=2\) तत्व होंगे।
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यदि सार्वत्रिक समुच्चय \(U=\{a,b,c,d,e\}\) और \(A=\{b,d\}\) है, तो (A') क्या होगा?
If the universal set is \(U=\{a,b,c,d,e\}\) and \(A=\{b,d\}\), what is (A')?
#sets
#universal set
#complement
#class 11
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A ({a,c,e})
B ({b,d})
C ({a,b,c,d,e})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({a,c,e})
Step 1
Concept
(A') contains the elements of (U) that are not in (A). Here those elements are (a,c,e).
Step 2
Why this answer is correct
The correct answer is A. ({a,c,e}). (A') contains the elements of (U) that are not in (A). Here those elements are (a,c,e).
Step 3
Exam Tip
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं होते। यहां वे तत्व (a,c,e) हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\) और \(A=\{1,3,5,7\}\) है, तो (A') में कितने तत्व हैं?
If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{1,3,5,7\}\), how many elements are in (A')?
#sets
#complement
#counting
#class 11
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
(A') contains (2,4,6,8), so it has (4) elements. A complement is always taken with respect to the given universal set.
Step 2
Why this answer is correct
The correct answer is C. (4). (A') contains (2,4,6,8), so it has (4) elements. A complement is always taken with respect to the given universal set.
Step 3
Exam Tip
(A') में (2,4,6,8) हैं, इसलिए इसमें (4) तत्व हैं। पूरक हमेशा दिए गए सार्वत्रिक समुच्चय के सापेक्ष होता है।
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एक कक्षा के (40) विद्यार्थियों में (18) गणित क्लब में और (15) विज्ञान क्लब में हैं। यदि (6) विद्यार्थी दोनों क्लब में हैं, तो किसी भी क्लब में नहीं होने वाले विद्यार्थी कितने हैं?
In a class of (40) students, (18) are in the maths club and (15) are in the science club. If (6) students are in both clubs, how many students are in neither club?
#sets
#universal set
#venn diagram
#class 11
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A (7)
B (13)
C (19)
D (27)
Explanation opens after your attempt
Step 1
Concept
(n\(A\cup B\)=18+15-6=27), so students outside are (40-27=13). In such questions, the universal set is the total group.
Step 2
Why this answer is correct
The correct answer is B. (13). (n\(A\cup B\)=18+15-6=27), so students outside are (40-27=13). In such questions, the universal set is the total group.
Step 3
Exam Tip
(n\(A\cup B\)=18+15-6=27), इसलिए बाहर के विद्यार्थी (40-27=13) हैं। ऐसे प्रश्नों में सार्वत्रिक समुच्चय कुल विद्यार्थियों का होता है।
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यदि (A) और (B), (U) के उपसमुच्चय हैं, तो (\(A\cup B\)') किसके बराबर है?
If (A) and (B) are subsets of (U), what is (\(A\cup B\)') equal to?
#sets
#de morgan law
#complement
#class 11
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A \(A'\cup B'\)
B \(A'\cap B'\)
C \(A\cap B\)
D \(A\cup B\)
Explanation opens after your attempt
Correct Answer
B. \(A'\cap B'\)
Step 1
Concept
By De Morgan's law, (\(A\cup B\)'=A'\cap B'). The complement of a union contains elements outside both sets.
Step 2
Why this answer is correct
The correct answer is B. \(A'\cap B'\). By De Morgan's law, (\(A\cup B\)'=A'\cap B'). The complement of a union contains elements outside both sets.
Step 3
Exam Tip
डी मॉर्गन नियम के अनुसार (\(A\cup B\)'=A'\cap B')। संघ के पूरक में केवल वे तत्व आते हैं जो दोनों से बाहर हों।
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यदि \(A\subseteq U\) है, तो ((A')') किसके बराबर होगा?
If \(A\subseteq U\), what will ((A')') be equal to?
#sets
#double complement
#universal set
#class 11
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A (A)
B (A')
C (U)
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The complement of the complement of a set is the set itself. This is called the double complement law.
Step 2
Why this answer is correct
The correct answer is A. (A). The complement of the complement of a set is the set itself. This is called the double complement law.
Step 3
Exam Tip
किसी समुच्चय के पूरक का पूरक वही समुच्चय होता है। इसे द्वि-पूरक नियम कहते हैं।
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यदि \(A\subseteq U\) और (A'=U) है, तो (A) क्या होगा?
If \(A\subseteq U\) and (A'=U), what is (A)?
#sets
#complement reasoning
#universal set
#class 11
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A (U)
B \(\varnothing\)
C (A')
D निश्चित नहीं / Not fixed
Explanation opens after your attempt
Correct Answer
B. \(\varnothing\)
Step 1
Concept
If the complement of (A) is the whole (U), then (A) cannot contain any element. Hence \(A=\varnothing\).
Step 2
Why this answer is correct
The correct answer is B. \(\varnothing\). If the complement of (A) is the whole (U), then (A) cannot contain any element. Hence \(A=\varnothing\).
Step 3
Exam Tip
यदि (A) का पूरक पूरा (U) है, तो (A) में कोई तत्व नहीं हो सकता। इसलिए \(A=\varnothing\) है।
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यदि (U) में (35) तत्व हैं और (A) में (12) तत्व हैं, तो (A') में कितने तत्व होंगे?
If (U) has (35) elements and (A) has (12) elements, how many elements will (A') have?
#sets
#complement
#cardinality
#class 11
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A (12)
B (23)
C (35)
D (47)
Explanation opens after your attempt
Step 1
Concept
(n(A')=n(U)-n(A)=35-12=23). To find a complement, subtract the set size from the total size.
Step 2
Why this answer is correct
The correct answer is B. (23). (n(A')=n(U)-n(A)=35-12=23). To find a complement, subtract the set size from the total size.
Step 3
Exam Tip
(n(A')=n(U)-n(A)=35-12=23)। पूरक की गणना में कुल से दिए हुए समुच्चय को घटाते हैं।
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यदि \(A={x:x\) संख्या (18) की धनात्मक भाजक है(}), तो (n(\mathcal{P}(A))) क्या होगा?
If \(A={x:x\) is a positive divisor of (18)(}), what is (n(\mathcal{P}(A)))?
#sets
#power set
#divisors
#class 11
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A (16)
B (32)
C (64)
D (18)
Explanation opens after your attempt
Step 1
Concept
The positive divisors of (18) are (1,2,3,6,9,18), so (n(A)=6). Hence (n(\mathcal{P}(A))=26 =64).
Step 2
Why this answer is correct
The correct answer is C. (64). The positive divisors of (18) are (1,2,3,6,9,18), so (n(A)=6). Hence (n(\mathcal{P}(A))=26 =64).
Step 3
Exam Tip
(18) के धनात्मक भाजक (1,2,3,6,9,18) हैं, इसलिए (n(A)=6)। अतः (n(\mathcal{P}(A))=26 =64)।
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यदि \(A=\{5\}\) है, तो (\mathcal{P}(A)) क्या होगा?
If \(A=\{5\}\), what is (\mathcal{P}(A))?
#sets
#singleton set
#power set
#class 11
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A \(\varnothing\)
B ({5})
C \({\varnothing,5}\)
D \({\varnothing,{5}}\)
Explanation opens after your attempt
Correct Answer
D. \({\varnothing,{5}}\)
Step 1
Concept
The subsets of a singleton set are \(\varnothing\) and the set itself. So ({5}), not (5), is an element of the power set.
Step 2
Why this answer is correct
The correct answer is D. \({\varnothing,{5}}\). The subsets of a singleton set are \(\varnothing\) and the set itself. So ({5}), not (5), is an element of the power set.
Step 3
Exam Tip
एकल समुच्चय के उपसमुच्चय \(\varnothing\) और वही समुच्चय होते हैं। इसलिए (5) नहीं, बल्कि ({5}) घात समुच्चय का तत्व है।
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यदि \({7}\in \mathcal{P}(A)\) है, तो कौन सा कथन निश्चित रूप से सही है?
If \({7}\in \mathcal{P}(A)\), which statement is definitely true?
#sets
#power set
#element subset
#class 11
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A \(7\in A\)
B \({7}\in A\)
C \(A=\{7\}\)
D \(7\notin A\)
Explanation opens after your attempt
Correct Answer
A. \(7\in A\)
Step 1
Concept
\({7}\in \mathcal{P}(A)\) means \({7}\subseteq A\). Therefore \(7\in A\) must be true.
Step 2
Why this answer is correct
The correct answer is A. \(7\in A\). \({7}\in \mathcal{P}(A)\) means \({7}\subseteq A\). Therefore \(7\in A\) must be true.
Step 3
Exam Tip
\({7}\in \mathcal{P}(A)\) का अर्थ है कि \({7}\subseteq A\)। इसलिए \(7\in A\) होना आवश्यक है।
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यदि \(A={\varnothing,1}\) है, तो निम्न में से कौन सा (\mathcal{P}(A)) का तत्व है?
If \(A={\varnothing,1}\), which of the following is an element of (\mathcal{P}(A))?
#sets
#empty set
#power set
#class 11
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A \(\varnothing\)
B \({\varnothing}\)
C ({1})
D उपरोक्त सभी / All of these
Explanation opens after your attempt
Correct Answer
D. उपरोक्त सभी / All of these
Step 1
Concept
\(\varnothing\), \({\varnothing}\), and ({1}) are all subsets of (A). Therefore all of them are elements of (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is D. उपरोक्त सभी / All of these. \(\varnothing\), \({\varnothing}\), and ({1}) are all subsets of (A). Therefore all of them are elements of (\mathcal{P}(A)).
Step 3
Exam Tip
\(\varnothing\), \({\varnothing}\), और ({1}) सभी (A) के उपसमुच्चय हैं। इसलिए ये सभी (\mathcal{P}(A)) के तत्व हैं।
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किसी भी समुच्चय (A) के लिए (\mathcal{P}(A)) में कौन से दो तत्व हमेशा होते हैं?
For any set (A), which two elements are always present in (\mathcal{P}(A))?
#sets
#power set
#basic property
#class 11
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A (A) और (U) / (A) and (U)
B \(\varnothing\) और (A) / \(\varnothing\) and (A)
C \(\varnothing\) और (U) / \(\varnothing\) and (U)
D केवल (A) / Only (A)
Explanation opens after your attempt
Correct Answer
B. \(\varnothing\) और (A) / \(\varnothing\) and (A)
Step 1
Concept
The empty set and the set itself are subsets of every set. Therefore both are always in its power set.
Step 2
Why this answer is correct
The correct answer is B. \(\varnothing\) और (A) / \(\varnothing\) and (A). The empty set and the set itself are subsets of every set. Therefore both are always in its power set.
Step 3
Exam Tip
हर समुच्चय का रिक्त समुच्चय और स्वयं वह समुच्चय उपसमुच्चय होते हैं। इसलिए दोनों हमेशा उसके घात समुच्चय में होते हैं।
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यदि (A) में (3) तत्व हैं, तो (A) के अरिक्त उचित उपसमुच्चयों की संख्या कितनी होगी?
If (A) has (3) elements, how many non-empty proper subsets of (A) are there?
#sets
#proper subsets
#non empty subsets
#class 11
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A (3)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^3=8\). Removing \(\varnothing\) and (A) gives (6) non-empty proper subsets.
Step 2
Why this answer is correct
The correct answer is B. (6). Total subsets are \(2^3=8\). Removing \(\varnothing\) and (A) gives (6) non-empty proper subsets.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^3=8\) हैं। अरिक्त उचित उपसमुच्चयों के लिए \(\varnothing\) और (A) हटाने पर (6) बचते हैं।
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यदि \(A=\{1,2,3,4\}\) है, तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जिनमें ठीक (2) तत्व हों?
If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) have exactly (2) elements?
#sets
#power set
#combination
#class 11
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The number of subsets with exactly (2) elements is \(\binom{4}{2}=6\). Inside a power set, subsets can also be counted by their size.
Step 2
Why this answer is correct
The correct answer is C. (6). The number of subsets with exactly (2) elements is \(\binom{4}{2}=6\). Inside a power set, subsets can also be counted by their size.
Step 3
Exam Tip
ठीक (2) तत्व वाले उपसमुच्चयों की संख्या \(\binom{4}{2}=6\) होती है। घात समुच्चय के अंदर उपसमुच्चयों को उनके आकार से भी गिना जा सकता है।
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यदि (A) में (5) तत्व हैं, तो (A) के ऐसे कितने उपसमुच्चय होंगे जिनमें एक निश्चित तत्व अवश्य हो?
If (A) has (5) elements, how many subsets of (A) will contain one fixed element?
#sets
#power set
#counting subsets
#class 11
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A (8)
B (10)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
After fixing one element, each of the remaining (4) elements may be chosen or not chosen. So the number is \(2^4=16\).
Step 2
Why this answer is correct
The correct answer is C. (16). After fixing one element, each of the remaining (4) elements may be chosen or not chosen. So the number is \(2^4=16\).
Step 3
Exam Tip
एक निश्चित तत्व रखने के बाद बाकी (4) तत्व चुने या छोड़े जा सकते हैं। इसलिए संख्या \(2^4=16\) होगी।
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यदि (A) में (6) तत्व हैं, तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें एक चुना हुआ तत्व न हो?
If (A) has (6) elements, how many subsets of (A) do not contain a selected element?
#sets
#power set
#subset counting
#class 11
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A (16)
B (32)
C (63)
D (64)
Explanation opens after your attempt
Step 1
Concept
After excluding the selected element, (5) elements remain. Their subsets are \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. (32). After excluding the selected element, (5) elements remain. Their subsets are \(2^5=32\).
Step 3
Exam Tip
चुना हुआ तत्व हटाने पर (5) तत्व बचते हैं। इनके उपसमुच्चय \(2^5=32\) होंगे।
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यदि \(A\subseteq U\), (n(U)=60), (n(A)=28), और (n(A')=x) है, तो (x) का मान क्या है?
If \(A\subseteq U\), (n(U)=60), (n(A)=28), and (n(A')=x), what is the value of (x)?
#sets
#universal set
#complement cardinality
#class 11
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A (28)
B (32)
C (60)
D (88)
Explanation opens after your attempt
Step 1
Concept
Elements of the complement are in (U) but not in (A). Therefore (x=60-28=32).
Step 2
Why this answer is correct
The correct answer is B. (32). Elements of the complement are in (U) but not in (A). Therefore (x=60-28=32).
Step 3
Exam Tip
पूरक के तत्व (U) में होते हैं पर (A) में नहीं होते। इसलिए (x=60-28=32) है।
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यदि \(U={x:x\in \mathbb{N}, x\leq 12}\) और \(A={x:x\) (3) का गुणज है(}), तो (A') क्या है?
If \(U={x:x\in \mathbb{N}, x\leq 12}\) and \(A={x:x\) is a multiple of (3)(}), what is (A')?
#sets
#universal set
#multiples
#class 11
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A ({3,6,9,12})
B ({1,2,4,5,7,8,10,11})
C ({1,3,5,7,9,11})
D ({2,4,6,8,10,12})
Explanation opens after your attempt
Correct Answer
B. ({1,2,4,5,7,8,10,11})
Step 1
Concept
(U) contains numbers from (1) to (12), and \(A=\{3,6,9,12\}\). Removing these gives (A').
Step 2
Why this answer is correct
The correct answer is B. ({1,2,4,5,7,8,10,11}). (U) contains numbers from (1) to (12), and \(A=\{3,6,9,12\}\). Removing these gives (A').
Step 3
Exam Tip
(U) में (1) से (12) तक संख्याएं हैं और \(A=\{3,6,9,12\}\) है। इन्हें हटाने पर (A') मिलता है।
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यदि \(A=\varnothing\) और सार्वत्रिक समुच्चय (U) है, तो (A') किसके बराबर होगा?
If \(A=\varnothing\) and the universal set is (U), what is (A') equal to?
#sets
#empty set
#universal set
#class 11
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A \(\varnothing\)
B (U)
C (\mathcal{P}(U))
D निश्चित नहीं / Not fixed
Explanation opens after your attempt
Step 1
Concept
The empty set has no elements, so all elements of (U) lie in its complement. Hence \(\varnothing'=U\).
Step 2
Why this answer is correct
The correct answer is B. (U). The empty set has no elements, so all elements of (U) lie in its complement. Hence \(\varnothing'=U\).
Step 3
Exam Tip
रिक्त समुच्चय में कोई तत्व नहीं होता, इसलिए (U) के सभी तत्व उसके पूरक में आते हैं। अतः \(\varnothing'=U\)।
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यदि \(A=\{1,2,3\}\) है, तो (\mathcal{P}(A)) में कितने अरिक्त तत्व होंगे?
If \(A=\{1,2,3\}\), how many non-empty elements are there in (\mathcal{P}(A))?
#sets
#power set
#non empty subsets
#class 11
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A (3)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^3=8\), and only \(\varnothing\) is empty. So non-empty subsets are (8-1=7).
Step 2
Why this answer is correct
The correct answer is C. (7). Total subsets are \(2^3=8\), and only \(\varnothing\) is empty. So non-empty subsets are (8-1=7).
Step 3
Exam Tip
कुल उपसमुच्चय \(2^3=8\) हैं और केवल \(\varnothing\) रिक्त है। इसलिए अरिक्त उपसमुच्चय (8-1=7) हैं।
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यदि \(A=\{m,n,o,p\}\) है, तो (\mathcal{P}(A)) में कितने एक-तत्वीय समुच्चय होंगे?
If \(A=\{m,n,o,p\}\), how many singleton sets will be in (\mathcal{P}(A))?
#sets
#power set
#singleton subsets
#class 11
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
A singleton subset is formed from each original element. From (4) elements, there will be (4) singleton subsets.
Step 2
Why this answer is correct
The correct answer is B. (4). A singleton subset is formed from each original element. From (4) elements, there will be (4) singleton subsets.
Step 3
Exam Tip
एक-तत्वीय उपसमुच्चय हर मूल तत्व से एक बनता है। (4) तत्वों से (4) singleton subsets बनेंगे।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ठीक (3) तत्व वाले समुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many sets in (\mathcal{P}(A)) have exactly (3) elements?
#sets
#power set
#combination
#class 11
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A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly (3) elements is \(\binom{5}{3}=10\). Each such choice is an element of (\mathcal{P}(A)).
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose exactly (3) elements is \(\binom{5}{3}=10\). Each such choice is an element of (\mathcal{P}(A)).
Step 3
Exam Tip
ठीक (3) तत्व चुनने के तरीके \(\binom{5}{3}=10\) हैं। ऐसे हर चयन से (\mathcal{P}(A)) का एक तत्व बनता है।
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यदि \(A=\{0,1\}\) है, तो (\mathcal{P}(A)) को सही रूप में कौन दिखाता है?
If \(A=\{0,1\}\), which correctly represents (\mathcal{P}(A))?
#sets
#power set
#representation
#class 11
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A \({\varnothing,{0},{1},{0,1}}\)
B ({0,1,{0,1}})
C ({{0},{1}})
D \({\varnothing,0,1}\)
Explanation opens after your attempt
Correct Answer
A. \({\varnothing,{0},{1},{0,1}}\)
Step 1
Concept
A two-element set has (4) subsets. In a power set, the elements are subsets, not the single numbers (0) and (1).
Step 2
Why this answer is correct
The correct answer is A. \({\varnothing,{0},{1},{0,1}}\). A two-element set has (4) subsets. In a power set, the elements are subsets, not the single numbers (0) and (1).
Step 3
Exam Tip
दो तत्वों वाले समुच्चय के (4) उपसमुच्चय होते हैं। घात समुच्चय में तत्व उपसमुच्चय होते हैं, अकेले (0) और (1) नहीं।
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यदि \(U=\{1,2,3,4,5,6\}\), \(A=\{1,2,3\}\) और \(B=\{3,4,5\}\) हैं, तो \(A'\cap B'\) क्या है?
If \(U=\{1,2,3,4,5,6\}\), \(A=\{1,2,3\}\), and \(B=\{3,4,5\}\), what is \(A'\cap B'\)?
#sets
#complement
#intersection
#class 11
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A ({6})
B ({1,2,3,4,5})
C ({3})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
(A'={4,5,6}) and (B'={1,2,6}). Their common part is ({6}).
Step 2
Why this answer is correct
The correct answer is A. ({6}). (A'={4,5,6}) and (B'={1,2,6}). Their common part is ({6}).
Step 3
Exam Tip
(A'={4,5,6}) और (B'={1,2,6}) हैं। इनका समान भाग ({6}) है।
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यदि \(U=\{a,b,c,d\}\), \(A=\{a,b\}\) और \(B=\{b,c\}\) हैं, तो (\(A\cap B\)') क्या होगा?
If \(U=\{a,b,c,d\}\), \(A=\{a,b\}\), and \(B=\{b,c\}\), what is (\(A\cap B\)')?
#sets
#universal set
#complement of intersection
#class 11
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A ({b})
B ({a,c,d})
C ({a,b,c})
D ({d})
Explanation opens after your attempt
Correct Answer
B. ({a,c,d})
Step 1
Concept
\(A\cap B={b}\), so its complement is all elements of (U) except (b). Hence ({a,c,d}) is correct.
Step 2
Why this answer is correct
The correct answer is B. ({a,c,d}). \(A\cap B={b}\), so its complement is all elements of (U) except (b). Hence ({a,c,d}) is correct.
Step 3
Exam Tip
\(A\cap B={b}\) है, इसलिए उसका पूरक (U) में (b) को छोड़कर बाकी तत्व हैं। अतः ({a,c,d}) सही है।
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यदि (n(A)=4) और (n(B)=3) है, तो (n(\mathcal{P}(A))) और (n(\mathcal{P}(B))) का अनुपात क्या होगा?
If (n(A)=4) and (n(B)=3), what is the ratio of (n(\mathcal{P}(A))) to (n(\mathcal{P}(B)))?
#sets
#power set
#ratio
#class 11
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A (2:1)
B (4:3)
C (8:3)
D (16:3)
Explanation opens after your attempt
Step 1
Concept
(n(\mathcal{P}(A))=16) and (n(\mathcal{P}(B))=8). The ratio is (16:8=2:1).
Step 2
Why this answer is correct
The correct answer is A. (2:1). (n(\mathcal{P}(A))=16) and (n(\mathcal{P}(B))=8). The ratio is (16:8=2:1).
Step 3
Exam Tip
(n(\mathcal{P}(A))=16) और (n(\mathcal{P}(B))=8) हैं। अनुपात (16:8=2:1) होगा।
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यदि \(A={x:x\) अंग्रेजी शब्द (LEVEL) के अलग-अलग अक्षर हैं(}), तो (\mathcal{P}(A)) में कितने तत्व होंगे?
If \(A={x:x\) is a distinct letter of the English word (LEVEL)(}), how many elements will (\mathcal{P}(A)) have?
#sets
#power set
#distinct elements
#class 11
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
The distinct letters of (LEVEL) are (L,E,V), so (n(A)=3). Therefore the power set has \(2^3=8\) elements.
Step 2
Why this answer is correct
The correct answer is A. (8). The distinct letters of (LEVEL) are (L,E,V), so (n(A)=3). Therefore the power set has \(2^3=8\) elements.
Step 3
Exam Tip
(LEVEL) के अलग-अलग अक्षर (L,E,V) हैं, इसलिए (n(A)=3)। अतः घात समुच्चय में \(2^3=8\) तत्व होंगे।
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यदि \(A=\{1,1,2,2,3\}\) को समुच्चय माना जाए, तो (n(\mathcal{P}(A))) कितना होगा?
If \(A=\{1,1,2,2,3\}\) is considered as a set, what is (n(\mathcal{P}(A)))?
#sets
#power set
#repeated elements
#class 11
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A (8)
B (16)
C (32)
D (5)
Explanation opens after your attempt
Step 1
Concept
Repeated elements are counted only once in a set, so \(A=\{1,2,3\}\). Hence (n(\mathcal{P}(A))=23 =8).
Step 2
Why this answer is correct
The correct answer is A. (8). Repeated elements are counted only once in a set, so \(A=\{1,2,3\}\). Hence (n(\mathcal{P}(A))=23 =8).
Step 3
Exam Tip
समुच्चय में दोहराए गए तत्व एक बार ही गिने जाते हैं, इसलिए \(A=\{1,2,3\}\)। इस कारण (n(\mathcal{P}(A))=23 =8)।
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यदि \(A=\{a,b,c,d,e,f\}\) है, तो (\mathcal{P}(A)) में ठीक (5) तत्व वाले उपसमुच्चयों की संख्या कितनी होगी?
If \(A=\{a,b,c,d,e,f\}\), how many subsets with exactly (5) elements are in (\mathcal{P}(A))?
#sets
#power set
#combination
#class 11
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A (5)
B (6)
C (12)
D (32)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose (5) elements from (6) is \(\binom{6}{5}=6\). Each choice is one element of the power set.
Step 2
Why this answer is correct
The correct answer is B. (6). The number of ways to choose (5) elements from (6) is \(\binom{6}{5}=6\). Each choice is one element of the power set.
Step 3
Exam Tip
(6) तत्वों में से (5) चुनने के तरीके \(\binom{6}{5}=6\) हैं। हर चयन घात समुच्चय का एक तत्व है।
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यदि (A) के कुल उपसमुच्चय (256) हैं, तो (A) में कितने तत्व हैं?
If (A) has (256) total subsets, how many elements does (A) have?
#sets
#power set
#total subsets
#class 11
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^n\), and \(256=2^8\). Therefore (A) has (8) elements.
Step 2
Why this answer is correct
The correct answer is C. (8). Total subsets are \(2^n\), and \(256=2^8\). Therefore (A) has (8) elements.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^n\) होते हैं और \(256=2^8\)। इसलिए (A) में (8) तत्व हैं।
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यदि \(A\subseteq U\), \(B\subseteq U\), और \(A\subseteq B\) है, तो पूरकों के लिए कौन सा संबंध सही है?
If \(A\subseteq U\), \(B\subseteq U\), and \(A\subseteq B\), which relation is correct for complements?
#sets
#complement
#subset relation
#class 11
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A \(A'\subseteq B'\)
B \(B'\subseteq A'\)
C (A'=B')
D \(A'\cap B'=\varnothing\)
Explanation opens after your attempt
Correct Answer
B. \(B'\subseteq A'\)
Step 1
Concept
If \(A\subseteq B\), then every element outside (B) is also outside (A). Hence \(B'\subseteq A'\).
Step 2
Why this answer is correct
The correct answer is B. \(B'\subseteq A'\). If \(A\subseteq B\), then every element outside (B) is also outside (A). Hence \(B'\subseteq A'\).
Step 3
Exam Tip
यदि \(A\subseteq B\) है, तो (B) के बाहर के तत्व (A) के बाहर भी होंगे। इसलिए \(B'\subseteq A'\) होता है।
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यदि \(A={x:x\in \mathbb{Z}, -2\leq x\leq 2}\) है, तो (\mathcal{P}(A)) में कितने तत्व होंगे?
If \(A={x:x\in \mathbb{Z}, -2\leq x\leq 2}\), how many elements will (\mathcal{P}(A)) have?
#sets
#power set
#integers
#class 11
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A (8)
B (16)
C (32)
D (64)
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Step 1
Concept
\(A=\{-2,-1,0,1,2\}\), so (n(A)=5). Therefore (n(\mathcal{P}(A))=25 =32).
Step 2
Why this answer is correct
The correct answer is C. (32). \(A=\{-2,-1,0,1,2\}\), so (n(A)=5). Therefore (n(\mathcal{P}(A))=25 =32).
Step 3
Exam Tip
\(A=\{-2,-1,0,1,2\}\), इसलिए (n(A)=5)। अतः (n(\mathcal{P}(A))=25 =32)।
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यदि \(A=\{1,2,3,4\}\) है, तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जिनमें (1) हो लेकिन (4) न हो?
If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) contain (1) but not (4)?
#sets
#power set
#conditional subsets
#class 11
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
Keeping (1) and excluding (4) are fixed. For the remaining (2,3), there are \(2^2=4\) choices.
Step 2
Why this answer is correct
The correct answer is B. (4). Keeping (1) and excluding (4) are fixed. For the remaining (2,3), there are \(2^2=4\) choices.
Step 3
Exam Tip
(1) को रखना और (4) को हटाना निश्चित है। बचे (2,3) के लिए \(2^2=4\) विकल्प हैं।
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यदि \(A=\{a,b,c,d,e\}\) है, तो (A) के ऐसे कितने उपसमुच्चय हैं जिनमें कम से कम (4) तत्व हों?
If \(A=\{a,b,c,d,e\}\), how many subsets of (A) contain at least (4) elements?
#sets
#power set
#at least subsets
#class 11
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A (5)
B (6)
C (10)
D (16)
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Step 1
Concept
At least (4) elements means (4) or (5) elements. The number is \(\binom{5}{4}+\binom{5}{5}=5+1=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). At least (4) elements means (4) or (5) elements. The number is \(\binom{5}{4}+\binom{5}{5}=5+1=6\).
Step 3
Exam Tip
कम से कम (4) तत्व का अर्थ (4) या (5) तत्व है। संख्या \(\binom{5}{4}+\binom{5}{5}=5+1=6\) है।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (2) और (5) दोनों हों?
If \(A=\{1,2,3,4,5,6\}\), how many subsets of (A) contain both (2) and (5)?
#sets
#power set
#subset counting
#class 11
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A (8)
B (16)
C (32)
D (64)
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Step 1
Concept
Keeping (2) and (5) is fixed, and the remaining (4) elements are free. Therefore the number of subsets is \(2^4=16\).
Step 2
Why this answer is correct
The correct answer is B. (16). Keeping (2) and (5) is fixed, and the remaining (4) elements are free. Therefore the number of subsets is \(2^4=16\).
Step 3
Exam Tip
(2) और (5) को रखना निश्चित है, बाकी (4) तत्व स्वतंत्र हैं। इसलिए उपसमुच्चय \(2^4=16\) होंगे।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (1) या (2) में से कोई भी न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets of (A) contain neither (1) nor (2)?
#sets
#power set
#restricted subsets
#class 11
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A (4)
B (8)
C (16)
D (32)
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Step 1
Concept
After excluding (1) and (2), the elements (3,4,5) remain. Their subsets are \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). After excluding (1) and (2), the elements (3,4,5) remain. Their subsets are \(2^3=8\).
Step 3
Exam Tip
(1) और (2) को हटाने के बाद (3,4,5) बचते हैं। इनके \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A=\{x,y,z\}\) है, तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जो (A) के उचित उपसमुच्चय भी हैं?
If \(A=\{x,y,z\}\), how many elements of (\mathcal{P}(A)) are also proper subsets of (A)?
#sets
#power set
#proper subset
#class 11
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A (3)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has (8) total subsets. For proper subsets, only (A) itself is removed, so (7) remain.
Step 2
Why this answer is correct
The correct answer is C. (7). (\mathcal{P}(A)) has (8) total subsets. For proper subsets, only (A) itself is removed, so (7) remain.
Step 3
Exam Tip
(\mathcal{P}(A)) में कुल (8) उपसमुच्चय हैं। उचित उपसमुच्चय के लिए केवल (A) को हटाते हैं, इसलिए (7) बचते हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,4,7\}\) और \(B=\{2,4,8\}\) हैं, तो (\(A\cup B\)') क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,4,7\}\), and \(B=\{2,4,8\}\), what is (\(A\cup B\)')?
#sets
#universal set
#union complement
#class 11
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A ({3,5,6,9})
B ({1,2,4,7,8})
C ({4})
D ({3,5,6})
Explanation opens after your attempt
Correct Answer
A. ({3,5,6,9})
Step 1
Concept
\(A\cup B={1,2,4,7,8}\). Removing these from (U) gives ({3,5,6,9}).
Step 2
Why this answer is correct
The correct answer is A. ({3,5,6,9}). \(A\cup B={1,2,4,7,8}\). Removing these from (U) gives ({3,5,6,9}).
Step 3
Exam Tip
\(A\cup B={1,2,4,7,8}\) है। (U) से इन्हें हटाने पर ({3,5,6,9}) मिलता है।
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यदि \(A=\{1,2,3\}\) और \(B=\{3,4\}\) हैं, तो (\mathcal{P}\(A\cup B\)) में कितने तत्व होंगे?
If \(A=\{1,2,3\}\) and \(B=\{3,4\}\), how many elements will (\mathcal{P}\(A\cup B\)) have?
#sets
#power set
#union
#class 11
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3,4}\), so it has (4) elements. Therefore its power set has \(2^4=16\) elements.
Step 2
Why this answer is correct
The correct answer is B. (16). \(A\cup B={1,2,3,4}\), so it has (4) elements. Therefore its power set has \(2^4=16\) elements.
Step 3
Exam Tip
\(A\cup B={1,2,3,4}\), इसलिए इसमें (4) तत्व हैं। अतः उसके घात समुच्चय में \(2^4=16\) तत्व होंगे।
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यदि \(U={x:x\in \mathbb{N}, x\leq 20}\) और \(A={x:x\) (20) का धनात्मक भाजक है(}) है, तो (n(A')) कितना होगा?
If \(U={x:x\in \mathbb{N}, x\leq 20}\) and \(A={x:x\) is a positive divisor of (20)(}), what is (n(A'))?
#sets
#universal set
#divisors complement
#class 11
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A (4)
B (6)
C (14)
D (20)
Explanation opens after your attempt
Step 1
Concept
The positive divisors of (20) are (1,2,4,5,10,20), so (n(A)=6). Thus (n(A')=20-6=14).
Step 2
Why this answer is correct
The correct answer is C. (14). The positive divisors of (20) are (1,2,4,5,10,20), so (n(A)=6). Thus (n(A')=20-6=14).
Step 3
Exam Tip
(20) के धनात्मक भाजक (1,2,4,5,10,20) हैं, इसलिए (n(A)=6)। (n(A')=20-6=14) होगा।
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