यदि \(A=\{2,4,6,8,10\}\) है, तो (\mathcal{P}(A)) के कितने तत्वों में सभी तत्व सम संख्या होंगे?
If \(A=\{2,4,6,8,10\}\), how many elements of (\mathcal{P}(A)) contain only even numbers?
#sets
#power-set
#even-numbers
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A (16)
B (31)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
All elements of (A) are even, so every subset in (\mathcal{P}(A)) contains only even numbers. The total number is \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is C. (32). All elements of (A) are even, so every subset in (\mathcal{P}(A)) contains only even numbers. The total number is \(2^5=32\).
Step 3
Exam Tip
(A) के सभी तत्व सम हैं, इसलिए (\mathcal{P}(A)) के हर उपसमुच्चय में केवल सम संख्याएँ होंगी। कुल संख्या \(2^5=32\) है।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (1) होगा, (2) नहीं होगा और ठीक (3) तत्व होंगे?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) contain (1), do not contain (2), and have exactly (3) elements?
#sets
#power-set
#restricted-subsets
#exact-size
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (2) is excluded, so choose (2) elements from (3,4,5). The count is \(\binom{3}{2}=3\).
Step 2
Why this answer is correct
The correct answer is B. (3). (1) is fixed and (2) is excluded, so choose (2) elements from (3,4,5). The count is \(\binom{3}{2}=3\).
Step 3
Exam Tip
(1) निश्चित है और (2) नहीं लेना है, इसलिए (3,4,5) में से (2) तत्व चुनने होंगे। संख्या \(\binom{3}{2}=3\) है।
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यदि (|A|=6), तो (\mathcal{P}(A)) में कितने तत्व ऐसे हैं जिनका आकार (2) या (4) है?
If (|A|=6), how many elements of (\mathcal{P}(A)) have size (2) or (4)?
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#power-set
#combination
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A (15)
B (30)
C (45)
D (60)
Explanation opens after your attempt
Step 1
Concept
The number is \(\binom{6}{2}+\binom{6}{4}=15+15=30\). Use combinations for size-based subset questions.
Step 2
Why this answer is correct
The correct answer is B. (30). The number is \(\binom{6}{2}+\binom{6}{4}=15+15=30\). Use combinations for size-based subset questions.
Step 3
Exam Tip
ऐसे उपसमुच्चयों की संख्या \(\binom{6}{2}+\binom{6}{4}=15+15=30\) है। आकार आधारित प्रश्नों में संयोजन लगाएँ।
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यदि \(U={x:x\in \mathbb{N}, x\le 24}\), (A) (2) के गुणजों का समुच्चय है और (B) (3) के गुणजों का समुच्चय है, तो (|\(A\cap B\)'|) क्या होगा?
If \(U={x:x\in \mathbb{N}, x\le 24}\), (A) is the set of multiples of (2) and (B) is the set of multiples of (3), what is (|\(A\cap B\)'|)?
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#universal-set
#intersection-complement
#multiples
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A (4)
B (12)
C (20)
D (24)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B\) contains the multiples of (6): (6,12,18,24), so it has (4) elements. The complement has (24-4=20) elements.
Step 2
Why this answer is correct
The correct answer is C. (20). \(A\cap B\) contains the multiples of (6): (6,12,18,24), so it has (4) elements. The complement has (24-4=20) elements.
Step 3
Exam Tip
\(A\cap B\) में (6) के गुणज (6,12,18,24) हैं, इसलिए इसमें (4) तत्व हैं। पूरक में (24-4=20) तत्व होंगे।
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\(यदि (U={1,2,3,4,5,6,7,8,9,10,11,12}) और (A={x:x\in U, x\) 4 से विभाज्य है}), तो (A') क्या है?
\(If (U={1,2,3,4,5,6,7,8,9,10,11,12}) and (A={x:x\in U, x\) is divisible by 4}), what is (A')?
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#universal-set
#complement
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A ({1,2,3,5,6,7,9,10,11})
B ({4,8,12})
C ({2,4,6,8,10,12})
D ({1,3,5,7,9,11})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3,5,6,7,9,10,11})
Step 1
Concept
The elements divisible by (4) are (4,8,12). The complement contains the remaining elements of (U).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,5,6,7,9,10,11}). The elements divisible by (4) are (4,8,12). The complement contains the remaining elements of (U).
Step 3
Exam Tip
(4) से विभाज्य तत्व (4,8,12) हैं। पूरक में (U) के बाकी तत्व आएँगे।
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यदि \(A={\emptyset,{a},b}\), तो निम्न में से कौन (\mathcal{P}(A)) का तत्व है?
If \(A={\emptyset,{a},b}\), which of the following is an element of (\mathcal{P}(A))?
#sets
#power-set
#nested-set
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A \({\emptyset,{a}}\)
B ({a,b})
C (a)
D ({{b}})
Explanation opens after your attempt
Correct Answer
A. \({\emptyset,{a}}\)
Step 1
Concept
\({\emptyset,{a}}\) is a subset of (A). A power set always contains subsets of the original set.
Step 2
Why this answer is correct
The correct answer is A. \({\emptyset,{a}}\). \({\emptyset,{a}}\) is a subset of (A). A power set always contains subsets of the original set.
Step 3
Exam Tip
\({\emptyset,{a}}\), (A) का उपसमुच्चय है। घात समुच्चय में हमेशा मूल समुच्चय के उपसमुच्चय आते हैं।
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यदि \(U=\mathbb{Z}\) और \(A={x:x\in \mathbb{Z}, -2\le x\le 3}\), तो (A') का सही वर्णन क्या है?
If \(U=\mathbb{Z}\) and \(A={x:x\in \mathbb{Z}, -2\le x\le 3}\), what is the correct description of (A')?
#sets
#universal-set
#integers
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A \({x:x\in \mathbb{Z}, x<-2\) या (x>3}) / \({x:x\in \mathbb{Z}, x<-2\) or (x>3})
B ({-2,-1,0,1,2,3})
C \({x:x\in \mathbb{Z}, -2<x<3}\)
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \({x:x\in \mathbb{Z}, x<-2\) या (x>3}) / \({x:x\in \mathbb{Z}, x<-2\) or (x>3})
Step 1
Concept
The complement contains integers outside the given interval. The endpoints (-2) and (3) are in (A), so they are not in the complement.
Step 2
Why this answer is correct
The correct answer is A. \({x:x\in \mathbb{Z}, x<-2\) या (x>3}) / \({x:x\in \mathbb{Z}, x<-2\) or (x>3}). The complement contains integers outside the given interval. The endpoints (-2) and (3) are in (A), so they are not in the complement.
Step 3
Exam Tip
पूरक में वे पूर्णांक होंगे जो दिए गए अंतराल के बाहर हैं। सीमा मान (-2) और (3) (A) में हैं इसलिए पूरक में नहीं होंगे।
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यदि (\mathcal{P}(A)) में (256) तत्व हैं, तो (\mathcal{P}(\mathcal{P}(A))) में कितने तत्व होंगे?
If (\mathcal{P}(A)) has (256) elements, how many elements are in (\mathcal{P}(\mathcal{P}(A)))?
#sets
#power-set
#iterated-power-set
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A (512)
B \(2^{256}\)
C \(256^2\)
D \(2^8\)
Explanation opens after your attempt
Correct Answer
B. \(2^{256}\)
Step 1
Concept
(\mathcal{P}(A)) itself is a set with (256) elements. Its power set therefore has \(2^{256}\) elements.
Step 2
Why this answer is correct
The correct answer is B. \(2^{256}\). (\mathcal{P}(A)) itself is a set with (256) elements. Its power set therefore has \(2^{256}\) elements.
Step 3
Exam Tip
(\mathcal{P}(A)) स्वयं (256) तत्वों वाला समुच्चय है। इसलिए उसके घात समुच्चय में \(2^{256}\) तत्व होंगे।
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यदि \(A=\{1,2,3,4,5,6\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (1) होगा और (6) नहीं होगा?
If \(A=\{1,2,3,4,5,6\}\), how many elements of (\mathcal{P}(A)) contain (1) and do not contain (6)?
#sets
#power-set
#restricted-subsets
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (6) is forbidden. The remaining four elements are free, so there are \(2^4=16\) subsets.
Step 2
Why this answer is correct
The correct answer is B. (16). (1) is fixed and (6) is forbidden. The remaining four elements are free, so there are \(2^4=16\) subsets.
Step 3
Exam Tip
(1) निश्चित है और (6) निषिद्ध है। शेष चार तत्व स्वतंत्र हैं, इसलिए \(2^4=16\) उपसमुच्चय होंगे।
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यदि \(U=\{a,b,c,d,e,g,h\}\), \(A=\{a,c,g\}\) और \(B=\{b,c,h\}\), तो (\(A\cup B\)') क्या है?
If \(U=\{a,b,c,d,e,g,h\}\), \(A=\{a,c,g\}\) and \(B=\{b,c,h\}\), what is (\(A\cup B\)')?
#sets
#universal-set
#union-complement
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A ({d,e})
B ({c})
C ({a,b,c,g,h})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({d,e})
Step 1
Concept
\(A\cup B={a,b,c,g,h}\). The elements left outside it in (U) are (d,e).
Step 2
Why this answer is correct
The correct answer is A. ({d,e}). \(A\cup B={a,b,c,g,h}\). The elements left outside it in (U) are (d,e).
Step 3
Exam Tip
\(A\cup B={a,b,c,g,h}\) है। (U) में इसके बाहर (d,e) बचते हैं।
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यदि \(A=\{p,q,r\}\), तो (\mathcal{P}(A)) के कितने अरिक्त उचित उपसमुच्चय हैं?
If \(A=\{p,q,r\}\), how many non-empty proper subsets are there in (\mathcal{P}(A))?
#sets
#power-set
#proper-subsets
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
There are \(2^3=8\) total subsets. Removing \(\emptyset\) and (A) leaves (6) non-empty proper subsets.
Step 2
Why this answer is correct
The correct answer is C. (6). There are \(2^3=8\) total subsets. Removing \(\emptyset\) and (A) leaves (6) non-empty proper subsets.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^3=8\) हैं। \(\emptyset\) और (A) हटाने पर (6) अरिक्त उचित उपसमुच्चय बचते हैं।
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यदि \(A\subseteq U\) और (A'=U), तो (A) के बारे में कौन सा कथन सही है?
If \(A\subseteq U\) and (A'=U), which statement about (A) is correct?
#sets
#universal-set
#empty-set
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A \(A=\emptyset\)
B (A=U)
C (A) में एक तत्व है / (A) has one element
D (A') खाली है / (A') is empty
Explanation opens after your attempt
Correct Answer
A. \(A=\emptyset\)
Step 1
Concept
Only the empty set has the whole (U) as its complement. Always remember the universal set in complement questions.
Step 2
Why this answer is correct
The correct answer is A. \(A=\emptyset\). Only the empty set has the whole (U) as its complement. Always remember the universal set in complement questions.
Step 3
Exam Tip
केवल रिक्त समुच्चय का पूरक पूरा (U) होता है। पूरक वाले प्रश्नों में सार्वत्रिक समुच्चय याद रखें।
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यदि \(A=\{1,{1},{1,2}\}\), तो निम्न में से कौन (A) का उपसमुच्चय है?
If \(A=\{1,{1},{1,2}\}\), which of the following is a subset of (A)?
#sets
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#element-vs-subset
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A ({1,{1,2}})
B ({1,2})
C ({{2}})
D ({2})
Explanation opens after your attempt
Correct Answer
A. ({1,{1,2}})
Step 1
Concept
Both (1) and ({1,2}) in the option are elements of (A). Treat ({1,2}) as one element of (A), not two separate elements.
Step 2
Why this answer is correct
The correct answer is A. ({1,{1,2}}). Both (1) and ({1,2}) in the option are elements of (A). Treat ({1,2}) as one element of (A), not two separate elements.
Step 3
Exam Tip
विकल्प में (1) और ({1,2}) दोनों (A) के तत्व हैं। ({1,2}) को (A) का तत्व समझें, दो अलग तत्व नहीं।
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यदि (|A|=7), तो (\mathcal{P}(A)) के कितने तत्वों में कम से कम (6) तत्व होंगे?
If (|A|=7), how many elements of (\mathcal{P}(A)) have at least (6) elements?
#sets
#power-set
#at-least-counting
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A (7)
B (8)
C (21)
D (128)
Explanation opens after your attempt
Step 1
Concept
At least (6) elements means (6) or (7) elements. The count is \(\binom{7}{6}+\binom{7}{7}=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). At least (6) elements means (6) or (7) elements. The count is \(\binom{7}{6}+\binom{7}{7}=8\).
Step 3
Exam Tip
कम से कम (6) तत्वों का अर्थ (6) या (7) तत्व है। संख्या \(\binom{7}{6}+\binom{7}{7}=8\) है।
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यदि \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,2,3,4\}\) और \(B=\{4,5,6\}\), तो \(A'\cap B\) क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,2,3,4\}\) and \(B=\{4,5,6\}\), what is \(A'\cap B\)?
#sets
#universal-set
#complement-intersection
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A ({5,6})
B ({4})
C ({7,8,9})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
A. ({5,6})
Step 1
Concept
(A'={5,6,7,8,9}). Intersecting it with (B) gives ({5,6}).
Step 2
Why this answer is correct
The correct answer is A. ({5,6}). (A'={5,6,7,8,9}). Intersecting it with (B) gives ({5,6}).
Step 3
Exam Tip
(A'={5,6,7,8,9}) है। इसे (B) से मिलाने पर ({5,6}) मिलता है।
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यदि \(A=\{0,1,2,3,4\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (0) होगा और उनका आकार विषम होगा?
If \(A=\{0,1,2,3,4\}\), how many elements of (\mathcal{P}(A)) contain (0) and have odd size?
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#power-set
#odd-cardinality
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
After fixing (0), an even number of the remaining four elements must be chosen. The count is \(\binom{4}{0}+\binom{4}{2}+\binom{4}{4}=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). After fixing (0), an even number of the remaining four elements must be chosen. The count is \(\binom{4}{0}+\binom{4}{2}+\binom{4}{4}=8\).
Step 3
Exam Tip
(0) को निश्चित रखने पर शेष चार तत्वों में से सम संख्या चुननी होगी। संख्या \(\binom{4}{0}+\binom{4}{2}+\binom{4}{4}=8\) है।
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यदि (U) सभी छोटे वर्णों का समुच्चय है और \(V=\{a,e,i,o,u\}\), तो (V') में कितने तत्व होंगे?
If (U) is the set of all lowercase alphabet letters and \(V=\{a,e,i,o,u\}\), how many elements are in (V')?
#sets
#universal-set
#complement-counting
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A (5)
B (21)
C (26)
D (31)
Explanation opens after your attempt
Step 1
Concept
(U) has (26) letters and (V) has (5) vowels. The complement has (26-5=21) consonants.
Step 2
Why this answer is correct
The correct answer is B. (21). (U) has (26) letters and (V) has (5) vowels. The complement has (26-5=21) consonants.
Step 3
Exam Tip
(U) में (26) वर्ण हैं और (V) में (5) स्वर हैं। पूरक में (26-5=21) व्यंजन होंगे।
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यदि \(A=\{a,b,c,d\}\), तो (\mathcal{P}(A)) के कितने तत्व ( {a,d}) से असंयुक्त नहीं हैं?
If \(A=\{a,b,c,d\}\), how many elements of (\mathcal{P}(A)) are not disjoint from ({a,d})?
#sets
#power-set
#disjoint-subsets
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
Subsets disjoint from ({a,d}) are formed only from (b,c), so there are \(2^2=4\). Hence not disjoint subsets are (16-4=12).
Step 2
Why this answer is correct
The correct answer is C. (12). Subsets disjoint from ({a,d}) are formed only from (b,c), so there are \(2^2=4\). Hence not disjoint subsets are (16-4=12).
Step 3
Exam Tip
({a,d}) से असंयुक्त उपसमुच्चय केवल (b,c) से बनेंगे, जो \(2^2=4\) हैं। इसलिए असंयुक्त नहीं होने वाले (16-4=12) हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6,8\}\) और \(B=\{1,2,3,4\}\), तो (\(A\cap B\)') क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6,8\}\) and \(B=\{1,2,3,4\}\), what is (\(A\cap B\)')?
#sets
#universal-set
#intersection-complement
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A ({1,3,5,6,7,8})
B ({2,4})
C ({5,6,7,8})
D ({1,2,3,4})
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,6,7,8})
Step 1
Concept
\(A\cap B={2,4}\). The elements outside it in (U) are ({1,3,5,6,7,8}).
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,6,7,8}). \(A\cap B={2,4}\). The elements outside it in (U) are ({1,3,5,6,7,8}).
Step 3
Exam Tip
\(A\cap B={2,4}\) है। (U) में इसके बाहर ({1,3,5,6,7,8}) हैं।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (2) और (3) में से कम से कम एक होगा?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) contain at least one of (2) and (3)?
#sets
#power-set
#at-least-one
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A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
There are (32) total subsets. Those containing neither (2) nor (3) are \(2^3=8\), so the answer is (24).
Step 2
Why this answer is correct
The correct answer is C. (24). There are (32) total subsets. Those containing neither (2) nor (3) are \(2^3=8\), so the answer is (24).
Step 3
Exam Tip
कुल उपसमुच्चय (32) हैं। (2) और (3) दोनों न लेने वाले \(2^3=8\) हैं, इसलिए उत्तर (24) है।
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यदि (\mathcal{P}(A)=\mathcal{P}(B)), तो कौन सा निष्कर्ष अनिवार्य है?
If (\mathcal{P}(A)=\mathcal{P}(B)), which conclusion is necessary?
#sets
#power-set
#equality
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A (A=B)
B \(A\cap B=\emptyset\)
C (A'=B)
D \(A\subset B'\)
Explanation opens after your attempt
Step 1
Concept
If power sets are equal, the original sets are also equal. This is because each original set is an element of its own power set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). If power sets are equal, the original sets are also equal. This is because each original set is an element of its own power set.
Step 3
Exam Tip
घात समुच्चय बराबर होने पर मूल समुच्चय भी बराबर होते हैं। क्योंकि मूल समुच्चय स्वयं अपने घात समुच्चय का तत्व होता है।
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\(यदि (U={x:x\in \mathbb{N}, x\le 20}) और (A={x:x\in U, x\) पूर्ण वर्ग है}), तो (|A'|) क्या है?
\(If (U={x:x\in \mathbb{N}, x\le 20}) and (A={x:x\in U, x\) is a perfect square}), what is (|A'|)?
#sets
#universal-set
#perfect-squares
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A (4)
B (14)
C (16)
D (20)
Explanation opens after your attempt
Step 1
Concept
The perfect squares up to (20) are (1,4,9,16), so (|A|=4). Thus (|A'|=20-4=16).
Step 2
Why this answer is correct
The correct answer is C. (16). The perfect squares up to (20) are (1,4,9,16), so (|A|=4). Thus (|A'|=20-4=16).
Step 3
Exam Tip
(20) तक पूर्ण वर्ग (1,4,9,16) हैं, इसलिए (|A|=4)। अतः (|A'|=20-4=16)।
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यदि \(A=\{r,s,t,u,v\}\), तो (\mathcal{P}(A)) के कितने तत्व ({r,s}) के अधिसमुच्चय हैं लेकिन (v) नहीं रखते?
If \(A=\{r,s,t,u,v\}\), how many elements of (\mathcal{P}(A)) are supersets of ({r,s}) but do not contain (v)?
#sets
#power-set
#superset-counting
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
(r,s) are fixed and (v) is excluded. The remaining (t,u) are free, so there are \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. (4). (r,s) are fixed and (v) is excluded. The remaining (t,u) are free, so there are \(2^2=4\) subsets.
Step 3
Exam Tip
(r,s) निश्चित हैं और (v) नहीं लेना है। शेष (t,u) स्वतंत्र हैं, इसलिए \(2^2=4\) उपसमुच्चय होंगे।
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यदि \(U=\{0,1,2,3,4,5,6,7,8,9\}\) और \(A={x:x\in U, x^2<25}\), तो (A') क्या है?
If \(U=\{0,1,2,3,4,5,6,7,8,9\}\) and \(A={x:x\in U, x^2<25}\), what is (A')?
#sets
#universal-set
#inequality
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A ({5,6,7,8,9})
B ({0,1,2,3,4})
C ({6,7,8,9})
D ({4,5})
Explanation opens after your attempt
Correct Answer
A. ({5,6,7,8,9})
Step 1
Concept
In (U), \(x^2<25\) gives (0,1,2,3,4). Therefore the complement is ({5,6,7,8,9}).
Step 2
Why this answer is correct
The correct answer is A. ({5,6,7,8,9}). In (U), \(x^2<25\) gives (0,1,2,3,4). Therefore the complement is ({5,6,7,8,9}).
Step 3
Exam Tip
\(x^2<25\) के लिए (U) में (0,1,2,3,4) आते हैं। इसलिए पूरक ({5,6,7,8,9}) है।
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यदि \(A=\{1,2,3,4\}\), तो ऐसे कितने क्रमित युग्म ((X,Y)) हैं जिनमें \(X\subseteq Y\subseteq A\) और (Y=A) है?
If \(A=\{1,2,3,4\}\), how many ordered pairs ((X,Y)) are there such that \(X\subseteq Y\subseteq A\) and (Y=A)?
#sets
#power-set
#ordered-pairs
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(Y=A) is fixed, so (X) can be any subset of (A). There are \(2^4=16\) choices for (X).
Step 2
Why this answer is correct
The correct answer is C. (16). (Y=A) is fixed, so (X) can be any subset of (A). There are \(2^4=16\) choices for (X).
Step 3
Exam Tip
(Y=A) निश्चित है, इसलिए (X), (A) का कोई भी उपसमुच्चय हो सकता है। (X) के \(2^4=16\) विकल्प हैं।
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यदि \(U=\{1,2,3,4,5,6,7\}\) और \(A=\{1,3,5,7\}\), तो (\mathcal{P}(A')) में कितने तत्व हैं?
If \(U=\{1,2,3,4,5,6,7\}\) and \(A=\{1,3,5,7\}\), how many elements are in (\mathcal{P}(A'))?
#sets
#power-set
#complement-power-set
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A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(A'={2,4,6}), which has (3) elements. Hence (|\mathcal{P}(A')|=23 =8).
Step 2
Why this answer is correct
The correct answer is B. (8). (A'={2,4,6}), which has (3) elements. Hence (|\mathcal{P}(A')|=23 =8).
Step 3
Exam Tip
(A'={2,4,6}) है, जिसमें (3) तत्व हैं। इसलिए (|\mathcal{P}(A')|=23 =8)।
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यदि (A) में (4) तत्व हैं और (B) में (5) तत्व हैं तथा \(A\subset B\), तो \(\mathcal{P}(B)\setminus\mathcal{P}(A)\) में कितने तत्व होंगे?
If (A) has (4) elements and (B) has (5) elements with \(A\subset B\), how many elements are in \(\mathcal{P}(B)\setminus\mathcal{P}(A)\)?
#sets
#power-set
#set-difference
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A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
(|\mathcal{P}(B)|=32) and (|\mathcal{P}(A)|=16). Since \(A\subset B\), the difference has (32-16=16) elements.
Step 2
Why this answer is correct
The correct answer is B. (16). (|\mathcal{P}(B)|=32) and (|\mathcal{P}(A)|=16). Since \(A\subset B\), the difference has (32-16=16) elements.
Step 3
Exam Tip
(|\mathcal{P}(B)|=32) और (|\mathcal{P}(A)|=16) है। क्योंकि \(A\subset B\), अंतर में (32-16=16) तत्व होंगे।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{1,2,3,4,5\}\) और \(B=\{2,4,6,8,10\}\), तो \(A'\cup B'\) में कितने तत्व हैं?
If \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{1,2,3,4,5\}\) and \(B=\{2,4,6,8,10\}\), how many elements are in \(A'\cup B'\)?
#sets
#universal-set
#de-morgan
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A (5)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
By De Morgan's law, (A'\cup B'=\(A\cap B\)'). Since \(A\cap B={2,4}\), the complement has (10-2=8) elements.
Step 2
Why this answer is correct
The correct answer is C. (8). By De Morgan's law, (A'\cup B'=\(A\cap B\)'). Since \(A\cap B={2,4}\), the complement has (10-2=8) elements.
Step 3
Exam Tip
डी मॉर्गन से (A'\cup B'=\(A\cap B\)') होता है। \(A\cap B={2,4}\), इसलिए पूरक में (10-2=8) तत्व हैं।
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यदि \(A={\emptyset,{\emptyset}}\), तो (|\mathcal{P}(A)|) क्या है?
If \(A={\emptyset,{\emptyset}}\), what is (|\mathcal{P}(A)|)?
#sets
#power-set
#empty-set
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A (2)
B (4)
C (8)
D (1)
Explanation opens after your attempt
Step 1
Concept
(A) has two distinct elements: \(\emptyset\) and \({\emptyset}\). Hence (|\mathcal{P}(A)|=22 =4).
Step 2
Why this answer is correct
The correct answer is B. (4). (A) has two distinct elements: \(\emptyset\) and \({\emptyset}\). Hence (|\mathcal{P}(A)|=22 =4).
Step 3
Exam Tip
(A) में दो अलग तत्व हैं: \(\emptyset\) और \({\emptyset}\)। इसलिए (|\mathcal{P}(A)|=22 =4)।
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यदि (U) सभी त्रिभुजों का समुच्चय है और (E) समबाहु त्रिभुजों का समुच्चय है, तो (E') क्या दर्शाता है?
If (U) is the set of all triangles and (E) is the set of equilateral triangles, what does (E') represent?
#sets
#universal-set
#interpretation
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A वे त्रिभुज जो समबाहु नहीं हैं / Triangles that are not equilateral
B सभी समबाहु त्रिभुज / All equilateral triangles
C सभी बहुभुज / All polygons
D केवल समकोण त्रिभुज / Only right triangles
Explanation opens after your attempt
Correct Answer
A. वे त्रिभुज जो समबाहु नहीं हैं / Triangles that are not equilateral
Step 1
Concept
Complement is always taken inside the universal set. So (E') is all triangles that are not equilateral.
Step 2
Why this answer is correct
The correct answer is A. वे त्रिभुज जो समबाहु नहीं हैं / Triangles that are not equilateral. Complement is always taken inside the universal set. So (E') is all triangles that are not equilateral.
Step 3
Exam Tip
पूरक हमेशा सार्वत्रिक समुच्चय के भीतर लिया जाता है। इसलिए (E') सभी ऐसे त्रिभुज हैं जो समबाहु नहीं हैं।
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यदि \(A=\{1,2,3,4,5,6,7\}\), तो (\mathcal{P}(A)) के कितने तत्वों में ठीक (5) तत्व नहीं होंगे?
If \(A=\{1,2,3,4,5,6,7\}\), how many elements of (\mathcal{P}(A)) do not have exactly (5) elements?
#sets
#power-set
#not-exactly
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A (21)
B (107)
C (121)
D (128)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^7=128\), and exactly (5)-element subsets are \(\binom{7}{5}=21\). So the answer is (128-21=107).
Step 2
Why this answer is correct
The correct answer is B. (107). Total subsets are \(2^7=128\), and exactly (5)-element subsets are \(\binom{7}{5}=21\). So the answer is (128-21=107).
Step 3
Exam Tip
कुल उपसमुच्चय \(2^7=128\) हैं और ठीक (5) तत्व वाले \(\binom{7}{5}=21\) हैं। इसलिए उत्तर (128-21=107) है।
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यदि \(U={1,2,3,\ldots,30}\) और (A) (2) या (3) से विभाज्य संख्याओं का समुच्चय है, तो (|A'|) क्या है?
If \(U={1,2,3,\ldots,30}\) and (A) is the set of numbers divisible by (2) or (3), what is (|A'|)?
#sets
#universal-set
#inclusion-exclusion
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A (10)
B (12)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
Numbers divisible by (2) or (3) are (15+10-5=20). Hence the complement has (30-20=10) numbers.
Step 2
Why this answer is correct
The correct answer is A. (10). Numbers divisible by (2) or (3) are (15+10-5=20). Hence the complement has (30-20=10) numbers.
Step 3
Exam Tip
(2) या (3) से विभाज्य संख्याएँ (15+10-5=20) हैं। इसलिए पूरक में (30-20=10) संख्याएँ हैं।
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यदि \(A=\{a,b,c,d,e\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (a) और (b) में से ठीक एक होगा और (e) होगा?
If \(A=\{a,b,c,d,e\}\), how many elements of (\mathcal{P}(A)) contain exactly one of (a) and (b) and also contain (e)?
#sets
#power-set
#exclusive-choice
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
There are (2) ways to choose exactly one of (a,b), and (e) is fixed. (c,d) are free, so \(2\times2^2=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). There are (2) ways to choose exactly one of (a,b), and (e) is fixed. (c,d) are free, so \(2\times2^2=8\).
Step 3
Exam Tip
(a,b) में से ठीक एक चुनने के (2) तरीके हैं और (e) निश्चित है। (c,d) स्वतंत्र हैं, इसलिए \(2\times2^2=8\)।
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यदि \(U={x:x\in \mathbb{Z}, -5\le x\le 5}\) और \(A={x:x\in U, |x|\le 2}\), तो (A') में कितने तत्व हैं?
If \(U={x:x\in \mathbb{Z}, -5\le x\le 5}\) and \(A={x:x\in U, |x|\le 2}\), how many elements are in (A')?
#sets
#universal-set
#absolute-value
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
(U) has (11) elements and \(A=\{-2,-1,0,1,2\}\) has (5) elements. So (A') has (6) elements.
Step 2
Why this answer is correct
The correct answer is C. (6). (U) has (11) elements and \(A=\{-2,-1,0,1,2\}\) has (5) elements. So (A') has (6) elements.
Step 3
Exam Tip
(U) में (11) तत्व हैं और \(A=\{-2,-1,0,1,2\}\) में (5) तत्व हैं। इसलिए (A') में (6) तत्व हैं।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) के कितने तत्वों का अधिकतम तत्व (3) है?
If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) have maximum element (3)?
#sets
#power-set
#maximum-element
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
Such a subset must contain (3) and must not contain (4). (1,2) are free, so there are \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. (4). Such a subset must contain (3) and must not contain (4). (1,2) are free, so there are \(2^2=4\) subsets.
Step 3
Exam Tip
ऐसे उपसमुच्चय में (3) होना चाहिए और (4) नहीं होना चाहिए। (1,2) स्वतंत्र हैं, इसलिए \(2^2=4\) उपसमुच्चय हैं।
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यदि \(A\subseteq B\subseteq U\), तो कौन सा कथन सदैव सत्य है?
If \(A\subseteq B\subseteq U\), which statement is always true?
#sets
#universal-set
#subset-complement
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A \(B'\subseteq A'\)
B \(A'\subseteq B'\)
C (A'=B')
D \(A\cap B' = B\)
Explanation opens after your attempt
Correct Answer
A. \(B'\subseteq A'\)
Step 1
Concept
Inclusion reverses when complements are taken. Thus \(A\subseteq B\) gives \(B'\subseteq A'\).
Step 2
Why this answer is correct
The correct answer is A. \(B'\subseteq A'\). Inclusion reverses when complements are taken. Thus \(A\subseteq B\) gives \(B'\subseteq A'\).
Step 3
Exam Tip
पूरक लेने पर समावेशन की दिशा उलट जाती है। इसलिए \(A\subseteq B\) से \(B'\subseteq A'\) मिलता है।
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यदि \(A=\{1,2,{3,4}\}\), तो (\mathcal{P}(A)) में कौन सा तत्व अवश्य होगा?
If \(A=\{1,2,{3,4}\}\), which element must be in (\mathcal{P}(A))?
#sets
#power-set
#nested-set
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A ({{3,4}})
B ({3,4})
C ({1,3})
D (3)
Explanation opens after your attempt
Correct Answer
A. ({{3,4}})
Step 1
Concept
({3,4}) is one element of (A), so its singleton subset ({{3,4}}) is in the power set.
Step 2
Why this answer is correct
The correct answer is A. ({{3,4}}). ({3,4}) is one element of (A), so its singleton subset ({{3,4}}) is in the power set.
Step 3
Exam Tip
({3,4}) (A) का एक तत्व है, इसलिए उसका एकल उपसमुच्चय ({{3,4}}) घात समुच्चय में होगा।
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यदि (|\mathcal{P}(A)|=32) और (|\mathcal{P}(B)|=8), तो (|\mathcal{P}\(A\times B\)|) क्या है?
If (|\mathcal{P}(A)|=32) and (|\mathcal{P}(B)|=8), what is (|\mathcal{P}\(A\times B\)|)?
#sets
#power-set
#cartesian-product
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A \(2^{15}\)
B \(2^8\)
C \(2^{10}\)
D \(2^{40}\)
Explanation opens after your attempt
Correct Answer
A. \(2^{15}\)
Step 1
Concept
(|A|=5) and (|B|=3), so \(|A\times B|=15\). Therefore its power set has \(2^{15}\) elements.
Step 2
Why this answer is correct
The correct answer is A. \(2^{15}\). (|A|=5) and (|B|=3), so \(|A\times B|=15\). Therefore its power set has \(2^{15}\) elements.
Step 3
Exam Tip
(|A|=5) और (|B|=3), इसलिए \(|A\times B|=15\) है। अतः उसके घात समुच्चय में \(2^{15}\) तत्व होंगे।
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\(यदि (U={1,2,3,4,5,6,7,8,9}) और (A={x:x\in U, x\) 2 से विभाज्य नहीं है}), तो (A') क्या है?
\(If (U={1,2,3,4,5,6,7,8,9}) and (A={x:x\in U, x\) is not divisible by 2}), what is (A')?
#sets
#universal-set
#even-odd
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A ({2,4,6,8})
B ({1,3,5,7,9})
C ({2,4,6,8,10})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({2,4,6,8})
Step 1
Concept
(A) contains the odd numbers of (U). Therefore the complement is the even numbers of (U): ({2,4,6,8}).
Step 2
Why this answer is correct
The correct answer is A. ({2,4,6,8}). (A) contains the odd numbers of (U). Therefore the complement is the even numbers of (U): ({2,4,6,8}).
Step 3
Exam Tip
(A) में (U) की विषम संख्याएँ हैं। इसलिए पूरक में (U) की सम संख्याएँ ({2,4,6,8}) होंगी।
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यदि \(A=\{1,2,3,4,5,6\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (1) से बड़ा कोई भी तत्व नहीं है?
If \(A=\{1,2,3,4,5,6\}\), how many elements of (\mathcal{P}(A)) contain no element greater than (1)?
#sets
#power-set
#condition-subsets
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A (1)
B (2)
C (6)
D (32)
Explanation opens after your attempt
Step 1
Concept
The only such subsets are \(\emptyset\) and ({1}). Do not forget to count the empty set in condition-based questions.
Step 2
Why this answer is correct
The correct answer is B. (2). The only such subsets are \(\emptyset\) and ({1}). Do not forget to count the empty set in condition-based questions.
Step 3
Exam Tip
ऐसे उपसमुच्चय केवल \(\emptyset\) और ({1}) हैं। शर्तों में खाली समुच्चय को भी गिनना न भूलें।
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यदि \(U=\mathbb{R}\) और (A=\(-\infty,2]\), तो (A') क्या है?
If \(U=\mathbb{R}\) and (A=\(-\infty,2]\), what is (A')?
#sets
#universal-set
#intervals
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A (\(2,\infty\))
B \([2,\infty\))
C (\(-\infty,2\))
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. (\(2,\infty\))
Step 1
Concept
(2) is included in (A), so it is not in the complement. In \(\mathbb{R}\), the complement is (\(2,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(2,\infty\)). (2) is included in (A), so it is not in the complement. In \(\mathbb{R}\), the complement is (\(2,\infty\)).
Step 3
Exam Tip
(2) (A) में शामिल है, इसलिए पूरक में (2) नहीं आएगा। \(\mathbb{R}\) में पूरक (\(2,\infty\)) है।
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यदि \(A=\{a,b,c,d,e,f\}\), तो (\mathcal{P}(A)) के कितने तत्वों में ठीक तीन तत्व होंगे और (a) नहीं होगा?
If \(A=\{a,b,c,d,e,f\}\), how many elements of (\mathcal{P}(A)) have exactly three elements and do not contain (a)?
#sets
#power-set
#exact-size
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A (10)
B (15)
C (20)
D (30)
Explanation opens after your attempt
Step 1
Concept
After excluding (a), (5) elements remain. Choosing exactly (3) of them gives \(\binom{5}{3}=10\) ways.
Step 2
Why this answer is correct
The correct answer is A. (10). After excluding (a), (5) elements remain. Choosing exactly (3) of them gives \(\binom{5}{3}=10\) ways.
Step 3
Exam Tip
(a) को हटाने पर (5) तत्व बचते हैं। उनमें से ठीक (3) चुनने के \(\binom{5}{3}=10\) तरीके हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), (A'={1,4,7}) और (B'={2,4,8}), तो \(A\cap B\) क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\), (A'={1,4,7}) and (B'={2,4,8}), what is \(A\cap B\)?
#sets
#universal-set
#de-morgan
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A ({3,5,6})
B ({4})
C ({1,2,4,7,8})
D ({3,4,5,6})
Explanation opens after your attempt
Correct Answer
A. ({3,5,6})
Step 1
Concept
(A\cap B=\(A'\cup B'\)'). Since \(A'\cup B'={1,2,4,7,8}\), the complement is ({3,5,6}).
Step 2
Why this answer is correct
The correct answer is A. ({3,5,6}). (A\cap B=\(A'\cup B'\)'). Since \(A'\cup B'={1,2,4,7,8}\), the complement is ({3,5,6}).
Step 3
Exam Tip
(A\cap B=\(A'\cup B'\)') है। \(A'\cup B'={1,2,4,7,8}\), इसलिए पूरक ({3,5,6}) है।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) के कितने तत्वों का योग सम है?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) have an even sum?
#sets
#power-set
#subset-sum
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A (8)
B (12)
C (16)
D (20)
Explanation opens after your attempt
Step 1
Concept
Since (A) has at least one odd number, subsets with even and odd sums are equal in number. Total subsets are (32), so even-sum subsets are (16).
Step 2
Why this answer is correct
The correct answer is C. (16). Since (A) has at least one odd number, subsets with even and odd sums are equal in number. Total subsets are (32), so even-sum subsets are (16).
Step 3
Exam Tip
(A) में कम से कम एक विषम संख्या है, इसलिए उपसमुच्चयों के योग सम और विषम बराबर संख्या में होंगे। कुल (32) हैं, अतः सम योग वाले (16) हैं।
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\(यदि (U={x:x\in \mathbb{N}, x\le 50}) और (A={x:x\in U, x\) 10 का गुणज है\(}), तो (|\mathcal{P}(A')|) क्या है\)?
\(If (U={x:x\in \mathbb{N}, x\le 50}) and (A={x:x\in U, x\) is a multiple of \(10}), what is (|\mathcal{P}(A')|)\)?
#sets
#power-set
#complement-cardinality
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A \(2^5\)
B \(2^{45}\)
C (45)
D \(2^{50}\)
Explanation opens after your attempt
Correct Answer
B. \(2^{45}\)
Step 1
Concept
The multiples of (10) are (10,20,30,40,50), so (|A|=5). (A') has (45) elements, hence (|\mathcal{P}(A')|=2^{45}).
Step 2
Why this answer is correct
The correct answer is B. \(2^{45}\). The multiples of (10) are (10,20,30,40,50), so (|A|=5). (A') has (45) elements, hence (|\mathcal{P}(A')|=2^{45}).
Step 3
Exam Tip
(10) के गुणज (10,20,30,40,50) हैं, इसलिए (|A|=5)। (A') में (45) तत्व हैं, अतः (|\mathcal{P}(A')|=2^{45})।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) के कितने तत्व ({1,2,3}) के उपसमुच्चय हैं लेकिन ({1}) के अधिसमुच्चय नहीं हैं?
If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) are subsets of ({1,2,3}) but not supersets of ({1})?
#sets
#power-set
#subset-superset
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The subset of ({1,2,3}) must not contain (1). Subsets formed only from (2,3) give \(2^2=4\) choices.
Step 2
Why this answer is correct
The correct answer is B. (4). The subset of ({1,2,3}) must not contain (1). Subsets formed only from (2,3) give \(2^2=4\) choices.
Step 3
Exam Tip
({1,2,3}) के उपसमुच्चय में (1) नहीं होना चाहिए। केवल (2,3) से बनने वाले \(2^2=4\) उपसमुच्चय मिलेंगे।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10,11,12\}\) और \(A=\{2,3,5,7,11\}\), तो (A') का सही वर्णन क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9,10,11,12\}\) and \(A=\{2,3,5,7,11\}\), what is the correct description of (A')?
#sets
#universal-set
#prime-composite
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A (U) में (1) और संयुक्त संख्याएँ / (1) and composite numbers in (U)
B (U) में अभाज्य संख्याएँ / Prime numbers in (U)
C (U) में केवल सम संख्याएँ / Only even numbers in (U)
D (U) में केवल विषम संख्याएँ / Only odd numbers in (U)
Explanation opens after your attempt
Correct Answer
A. (U) में (1) और संयुक्त संख्याएँ / (1) and composite numbers in (U)
Step 1
Concept
(A) contains the prime numbers of (U). Therefore the complement contains (1) and composite numbers.
Step 2
Why this answer is correct
The correct answer is A. (U) में (1) और संयुक्त संख्याएँ / (1) and composite numbers in (U). (A) contains the prime numbers of (U). Therefore the complement contains (1) and composite numbers.
Step 3
Exam Tip
(A) में (U) की अभाज्य संख्याएँ हैं। इसलिए पूरक में (1) और संयुक्त संख्याएँ होंगी।
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यदि \(A=\{x,y,z\}\), तो \(\mathcal{P}(\mathcal{P}(A))\) में कितने एकतत्वीय समुच्चय होंगे?
If \(A=\{x,y,z\}\), then how many singleton sets are there in \(\mathcal{P}(\mathcal{P}(A))\)?
#sets
#power-set
#iterated-power-set
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A (3)
B (8)
C (16)
D (256)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^3=8\) elements. The power set of an (8)-element set has (8) singleton subsets.
Step 2
Why this answer is correct
The correct answer is B. (8). (\mathcal{P}(A)) has \(2^3=8\) elements. The power set of an (8)-element set has (8) singleton subsets.
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^3=8\) तत्व हैं। किसी (8)-तत्वीय समुच्चय के घात समुच्चय में (8) एकल उपसमुच्चय होते हैं।
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यदि \(U=\{a,b,c,d,e,f,g,h\}\), \(A=\{a,b,c,d\}\) और \(B=\{c,d,e,f\}\), तो \(A'\triangle B'\) में कितने तत्व हैं?
If \(U=\{a,b,c,d,e,f,g,h\}\), \(A=\{a,b,c,d\}\) and \(B=\{c,d,e,f\}\), how many elements are in \(A'\triangle B'\)?
#sets
#universal-set
#symmetric-difference
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(A'={e,f,g,h}) and (B'={a,b,g,h}). Elements in exactly one are (a,b,e,f), so there are (4).
Step 2
Why this answer is correct
The correct answer is B. (4). (A'={e,f,g,h}) and (B'={a,b,g,h}). Elements in exactly one are (a,b,e,f), so there are (4).
Step 3
Exam Tip
(A'={e,f,g,h}) और (B'={a,b,g,h}) हैं। केवल एक में आने वाले तत्व (a,b,e,f) हैं, इसलिए (4) तत्व हैं।
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यदि \(A=\{1,2,3,4,5,6\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (1,2,3) में से कम से कम दो तत्व होंगे?
If \(A=\{1,2,3,4,5,6\}\), how many elements of (\mathcal{P}(A)) contain at least two elements from (1,2,3)?
#sets
#power-set
#at-least-two
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A (16)
B (24)
C (32)
D (48)
Explanation opens after your attempt
Step 1
Concept
Choosing at least two from (1,2,3) gives \(\binom{3}{2}+\binom{3}{3}=4\) ways. The remaining three elements are free, so the count is \(4\times2^3=32\).
Step 2
Why this answer is correct
The correct answer is C. (32). Choosing at least two from (1,2,3) gives \(\binom{3}{2}+\binom{3}{3}=4\) ways. The remaining three elements are free, so the count is \(4\times2^3=32\).
Step 3
Exam Tip
(1,2,3) में से कम से कम दो चुनने के \(\binom{3}{2}+\binom{3}{3}=4\) तरीके हैं। शेष तीन तत्व स्वतंत्र हैं, इसलिए \(4\times2^3=32\) है।
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