यदि \(A=\{1,2,3,4\}\) है और \(B\subseteq A\) में ठीक दो तत्व हैं, तो ऐसे कितने (B), (\mathcal{P}(A)) के तत्व होंगे?
If \(A=\{1,2,3,4\}\) and \(B\subseteq A\) has exactly two elements, how many such (B) will be elements of (\mathcal{P}(A))?
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#power-set
#combination
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A (6)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
The number of two-element subsets is \(\binom{4}{2}=6\). In exams, remember elements of (\mathcal{P}(A)) are subsets.
Step 2
Why this answer is correct
The correct answer is A. (6). The number of two-element subsets is \(\binom{4}{2}=6\). In exams, remember elements of (\mathcal{P}(A)) are subsets.
Step 3
Exam Tip
दो तत्वों वाले उपसमुच्चयों की संख्या \(\binom{4}{2}=6\) है। परीक्षा में (\mathcal{P}(A)) के तत्व उपसमुच्चय होते हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{3,4,5,6\}\) हैं, तो (|\mathcal{P}\(A\cap B\)|) का मान क्या होगा?
If \(A=\{1,2,3,4\}\) and \(B=\{3,4,5,6\}\), what is the value of (|\mathcal{P}\(A\cap B\)|)?
#sets
#power-set
#intersection
#cardinality
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B={3,4}\), so its power set has \(2^2=4\) elements. First find the intersection, then apply the power set formula.
Step 2
Why this answer is correct
The correct answer is B. (4). \(A\cap B={3,4}\), so its power set has \(2^2=4\) elements. First find the intersection, then apply the power set formula.
Step 3
Exam Tip
\(A\cap B={3,4}\) है, इसलिए इसके घात समुच्चय में \(2^2=4\) तत्व होंगे। पहले प्रतिच्छेद निकालें, फिर घात समुच्चय की संख्या लगाएँ।
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यदि (|\mathcal{P}(A)|=128) है, तो (|A|) का मान क्या होगा?
If (|\mathcal{P}(A)|=128), what is the value of (|A|)?
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#power-set
#cardinality
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A (6)
B (7)
C (8)
D (14)
Explanation opens after your attempt
Step 1
Concept
If (|A|=n), then (|\mathcal{P}(A)|=2^n). Since \(128=2^7\), (|A|=7).
Step 2
Why this answer is correct
The correct answer is B. (7). If (|A|=n), then (|\mathcal{P}(A)|=2^n). Since \(128=2^7\), (|A|=7).
Step 3
Exam Tip
यदि (|A|=n), तो (|\mathcal{P}(A)|=2^n) होता है। \(128=2^7\), इसलिए (|A|=7)।
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यदि \(U={x:x\in \mathbb{N}, x\le 15}\) और \(A={x:x\in U, x\) अभाज्य नहीं है(}), तो (A') क्या होगा?
If \(U={x:x\in \mathbb{N}, x\le 15}\) and \(A={x:x\in U, x\) is not prime(}), what will (A') be?
#sets
#universal-set
#complement
#prime-numbers
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A ({2,3,5,7,11,13})
B ({1,4,6,8,9,10,12,14,15})
C ({1,2,3,5,7,11,13})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({2,3,5,7,11,13})
Step 1
Concept
(A) contains the numbers that are not prime, so (A') contains the prime numbers of (U). Remember that (1) is not prime.
Step 2
Why this answer is correct
The correct answer is A. ({2,3,5,7,11,13}). (A) contains the numbers that are not prime, so (A') contains the prime numbers of (U). Remember that (1) is not prime.
Step 3
Exam Tip
(A) में अभाज्य नहीं होने वाली संख्याएँ हैं, इसलिए (A') में (U) की अभाज्य संख्याएँ आएँगी। ध्यान रखें कि (1) अभाज्य नहीं होता।
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यदि \(U=\{1,2,3,4,5,6,7,8,9\}\) और \(A=\{2,4,6,8\}\) है, तो (A') क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9\}\) and \(A=\{2,4,6,8\}\), what is (A')?
#sets
#universal-set
#complement
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A ({1,3,5,7,9})
B ({2,4,6,8})
C ({1,2,3,4,5,6,7,8,9})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,7,9})
Step 1
Concept
The complement (A') contains elements of (U) that are not in (A). Always check the universal set first.
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,7,9}). The complement (A') contains elements of (U) that are not in (A). Always check the universal set first.
Step 3
Exam Tip
पूरक (A') में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। सार्वत्रिक समुच्चय हमेशा पहले देखें।
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यदि \(A=\{a,b,c,d,e\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (a) होगा लेकिन (b) और (c) दोनों नहीं होंगे?
If \(A=\{a,b,c,d,e\}\), how many elements of (\mathcal{P}(A)) contain (a) but do not contain both (b) and (c)?
#sets
#power-set
#restricted-subsets
#counting
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A (12)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
(a) is fixed and (d,e) are free. For (b,c), remove the one case where both are chosen, so the count is \(3\times 2^2=12\).
Step 2
Why this answer is correct
The correct answer is A. (12). (a) is fixed and (d,e) are free. For (b,c), remove the one case where both are chosen, so the count is \(3\times 2^2=12\).
Step 3
Exam Tip
(a) निश्चित है और (d,e) स्वतंत्र हैं। (b,c) के लिए कुल (4) विकल्पों में से दोनों साथ वाला विकल्प हटेगा, इसलिए \(3\times 2^2=12\)।
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यदि \(A={\emptyset,{1},2}\) है, तो निम्न में से कौन (\mathcal{P}(A)) का तत्व है?
If \(A={\emptyset,{1},2}\), which of the following is an element of (\mathcal{P}(A))?
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#power-set
#empty-set
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A \({\emptyset,2}\)
B (1)
C ({1,2})
D ({{2}})
Explanation opens after your attempt
Correct Answer
A. \({\emptyset,2}\)
Step 1
Concept
\({\emptyset,2}\) is a subset of (A), so it is an element of the power set. (1) and ({2}) are not elements of (A).
Step 2
Why this answer is correct
The correct answer is A. \({\emptyset,2}\). \({\emptyset,2}\) is a subset of (A), so it is an element of the power set. (1) and ({2}) are not elements of (A).
Step 3
Exam Tip
\({\emptyset,2}\), (A) का उपसमुच्चय है, इसलिए यह घात समुच्चय का तत्व है। (1) और ({2}), (A) के तत्व नहीं हैं।
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यदि \(U={x:x\in \mathbb{Z}, -3\le x\le 4}\) और \(A={x:x\in U, x^2<4}\) है, तो (A') में कितने तत्व हैं?
If \(U={x:x\in \mathbb{Z}, -3\le x\le 4}\) and \(A={x:x\in U, x^2<4}\), how many elements are in (A')?
#sets
#universal-set
#integer-set
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
Here \(U=\{-3,-2,-1,0,1,2,3,4\}\) and \(A=\{-1,0,1\}\). Hence (A') has (5) elements.
Step 2
Why this answer is correct
The correct answer is C. (5). Here \(U=\{-3,-2,-1,0,1,2,3,4\}\) and \(A=\{-1,0,1\}\). Hence (A') has (5) elements.
Step 3
Exam Tip
\(U=\{-3,-2,-1,0,1,2,3,4\}\) और \(A=\{-1,0,1\}\) है। इसलिए (A') में (5) तत्व हैं।
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यदि (A) में (5) तत्व हैं, तो (\mathcal{P}(A)) के कितने तत्व स्वयं एकल समुच्चय होंगे?
If (A) has (5) elements, how many elements of (\mathcal{P}(A)) will be singleton sets?
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#power-set
#singleton
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A (4)
B (5)
C (10)
D (32)
Explanation opens after your attempt
Step 1
Concept
Singleton elements of (\mathcal{P}(A)) are one-element subsets of (A). Their number is \(\binom{5}{1}=5\).
Step 2
Why this answer is correct
The correct answer is B. (5). Singleton elements of (\mathcal{P}(A)) are one-element subsets of (A). Their number is \(\binom{5}{1}=5\).
Step 3
Exam Tip
(\mathcal{P}(A)) के एकल तत्व वे उपसमुच्चय हैं जिनमें (1) तत्व है। उनकी संख्या \(\binom{5}{1}=5\) है।
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यदि \(A=\{a,b,c\}\), तो (\mathcal{P}(A)) में कौन सा तत्व नहीं होगा?
If \(A=\{a,b,c\}\), which will not be an element of (\mathcal{P}(A))?
#sets
#power-set
#common-mistake
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A ({a,b})
B \(\emptyset\)
C ({c})
D (a)
Explanation opens after your attempt
Step 1
Concept
Every element of (\mathcal{P}(A)) must be a subset of (A). (a) is an element, not a subset.
Step 2
Why this answer is correct
The correct answer is D. (a). Every element of (\mathcal{P}(A)) must be a subset of (A). (a) is an element, not a subset.
Step 3
Exam Tip
(\mathcal{P}(A)) का हर तत्व (A) का उपसमुच्चय होना चाहिए। (a) एक तत्व है, उपसमुच्चय नहीं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{2,3,5,7\}\), तो (A') किस प्रकार का समुच्चय है?
If \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{2,3,5,7\}\), what type of set is (A')?
#sets
#universal-set
#prime-numbers
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A अभाज्य संख्याएँ / Prime numbers
B संयुक्त संख्याएँ और (1) / Composite numbers and (1)
C सम संख्याएँ / Even numbers
D विषम संख्याएँ / Odd numbers
Explanation opens after your attempt
Correct Answer
B. संयुक्त संख्याएँ और (1) / Composite numbers and (1)
Step 1
Concept
(A) is the set of prime numbers present in (U). So the complement contains (1) and composite numbers.
Step 2
Why this answer is correct
The correct answer is B. संयुक्त संख्याएँ और (1) / Composite numbers and (1). (A) is the set of prime numbers present in (U). So the complement contains (1) and composite numbers.
Step 3
Exam Tip
(A), (U) में उपस्थित अभाज्य संख्याओं का समुच्चय है। इसलिए पूरक में (1) और संयुक्त संख्याएँ रहेंगी।
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यदि (\mathcal{P}(A)={\emptyset,{p},{q},{p,q}}), तो (A) क्या है?
If (\mathcal{P}(A)={\emptyset,{p},{q},{p,q}}), what is (A)?
#sets
#power-set
#reconstruction
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A \({\emptyset,p,q}\)
B ({p,q})
C ({{p},{q}})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
B. ({p,q})
Step 1
Concept
The largest subset shown in the power set is ({p,q}), which is the original set. Hence \(A=\{p,q\}\).
Step 2
Why this answer is correct
The correct answer is B. ({p,q}). The largest subset shown in the power set is ({p,q}), which is the original set. Hence \(A=\{p,q\}\).
Step 3
Exam Tip
दिए गए घात समुच्चय में मूल समुच्चय का सबसे बड़ा उपसमुच्चय ({p,q}) है। इसलिए \(A=\{p,q\}\)।
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यदि \(A\subseteq B\) और \(B\subseteq U\), तो \(B'\subseteq A'\) क्यों सत्य है?
If \(A\subseteq B\) and \(B\subseteq U\), why is \(B'\subseteq A'\) true?
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#universal-set
#subset-reasoning
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A बड़ा समुच्चय छोटा पूरक देता है / A larger set gives a smaller complement
B हर पूरक खाली होता है / Every complement is empty
C (A) और (B) बराबर हैं / (A) and (B) are equal
D पूरक में क्रम नहीं बदलता / Complement does not reverse order
Explanation opens after your attempt
Correct Answer
A. बड़ा समुच्चय छोटा पूरक देता है / A larger set gives a smaller complement
Step 1
Concept
When \(A\subseteq B\), everything outside (B) is also outside (A). Inclusion reverses under complement.
Step 2
Why this answer is correct
The correct answer is A. बड़ा समुच्चय छोटा पूरक देता है / A larger set gives a smaller complement. When \(A\subseteq B\), everything outside (B) is also outside (A). Inclusion reverses under complement.
Step 3
Exam Tip
जब \(A\subseteq B\), तो (B) के बाहर की हर वस्तु (A) के बाहर भी होगी। पूरक लेने पर समावेशन उलट जाता है।
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यदि (|A|=4), तो (\mathcal{P}(A)) के कितने तत्वों में कम से कम (3) तत्व होंगे?
If (|A|=4), how many elements of (\mathcal{P}(A)) have at least (3) elements?
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#power-set
#subset-counting
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A (4)
B (5)
C (8)
D (11)
Explanation opens after your attempt
Step 1
Concept
Subsets with at least (3) elements have (3) or (4) elements. The number is \(\binom{4}{3}+\binom{4}{4}=5\).
Step 2
Why this answer is correct
The correct answer is B. (5). Subsets with at least (3) elements have (3) or (4) elements. The number is \(\binom{4}{3}+\binom{4}{4}=5\).
Step 3
Exam Tip
कम से कम (3) तत्वों वाले उपसमुच्चय (3) या (4) तत्वों के होंगे। संख्या \(\binom{4}{3}+\binom{4}{4}=5\) है।
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यदि \(U=\{a,b,c,d,e\}\), \(A=\{a,c,e\}\) और \(B=\{b,c,d\}\), तो (\(A\cup B\)') क्या है?
If \(U=\{a,b,c,d,e\}\), \(A=\{a,c,e\}\) and \(B=\{b,c,d\}\), what is (\(A\cup B\)')?
#sets
#universal-set
#union-complement
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A \(\emptyset\)
B ({c})
C ({a,b,d,e})
D ({a,b,c,d,e})
Explanation opens after your attempt
Correct Answer
A. \(\emptyset\)
Step 1
Concept
\(A\cup B={a,b,c,d,e}=U\). Therefore its complement is the empty set.
Step 2
Why this answer is correct
The correct answer is A. \(\emptyset\). \(A\cup B={a,b,c,d,e}=U\). Therefore its complement is the empty set.
Step 3
Exam Tip
\(A\cup B={a,b,c,d,e}=U\) है। इसलिए इसका पूरक खाली समुच्चय है।
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यदि \(A=\{1,{2}\}\), तो (\mathcal{P}(A)) में कौन सा अवश्य होगा?
If \(A=\{1,{2}\}\), which must be in (\mathcal{P}(A))?
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#power-set
#nested-set
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A ({2})
B ({{2}})
C (2)
D ({1,2})
Explanation opens after your attempt
Correct Answer
B. ({{2}})
Step 1
Concept
({2}) is an element of (A), so its singleton subset ({{2}}) is in the power set. Keep a clear difference between a set and its element.
Step 2
Why this answer is correct
The correct answer is B. ({{2}}). ({2}) is an element of (A), so its singleton subset ({{2}}) is in the power set. Keep a clear difference between a set and its element.
Step 3
Exam Tip
({2}), (A) का एक तत्व है, इसलिए उसका एकल उपसमुच्चय ({{2}}) घात समुच्चय में होगा। सेट और उसके तत्व में अंतर रखें।
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यदि सार्वत्रिक समुच्चय \(U=\mathbb{R}\) और \(A={x:x^2=4}\) है, तो (A') क्या दर्शाता है?
If the universal set is \(U=\mathbb{R}\) and \(A={x:x^2=4}\), what does (A') represent?
#sets
#universal-set
#real-numbers
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A सभी वास्तविक संख्याएँ सिवाय (-2) और (2) / All real numbers except (-2) and (2)
B केवल (-2) और (2) / Only (-2) and (2)
C सभी पूर्णांक / All integers
D खाली समुच्चय / Empty set
Explanation opens after your attempt
Correct Answer
A. सभी वास्तविक संख्याएँ सिवाय (-2) और (2) / All real numbers except (-2) and (2)
Step 1
Concept
The solutions of \(x^2=4\) are (-2) and (2). In \(\mathbb{R}\), the complement is all real numbers except these two.
Step 2
Why this answer is correct
The correct answer is A. सभी वास्तविक संख्याएँ सिवाय (-2) और (2) / All real numbers except (-2) and (2). The solutions of \(x^2=4\) are (-2) and (2). In \(\mathbb{R}\), the complement is all real numbers except these two.
Step 3
Exam Tip
\(x^2=4\) के हल (-2) और (2) हैं। \(\mathbb{R}\) में पूरक इन दोनों को छोड़कर सभी वास्तविक संख्याएँ हैं।
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यदि (A) एक रिक्त समुच्चय है, तो (\mathcal{P}(\mathcal{P}(A))) में कितने तत्व होंगे?
If (A) is an empty set, how many elements are in (\mathcal{P}(\mathcal{P}(A)))?
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#power-set
#empty-set
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A (1)
B (2)
C (4)
D (0)
Explanation opens after your attempt
Step 1
Concept
If \(A=\emptyset\), then (\mathcal{P}(A)={\emptyset}). Hence (\mathcal{P}(\mathcal{P}(A))) has \(2^1=2\) elements.
Step 2
Why this answer is correct
The correct answer is B. (2). If \(A=\emptyset\), then (\mathcal{P}(A)={\emptyset}). Hence (\mathcal{P}(\mathcal{P}(A))) has \(2^1=2\) elements.
Step 3
Exam Tip
यदि \(A=\emptyset\), तो (\mathcal{P}(A)={\emptyset}) है। इसलिए (\mathcal{P}(\mathcal{P}(A))) में \(2^1=2\) तत्व हैं।
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यदि \(A=\{1,2,3\}\), तो (\mathcal{P}(A)) के कितने तत्व (1) को शामिल करते हैं?
If \(A=\{1,2,3\}\), how many elements of (\mathcal{P}(A)) contain (1)?
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#power-set
#fixed-element
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
Fix (1), and the remaining two elements may be chosen or not chosen in \(2^2=4\) ways. With a fixed element, the rest are independent.
Step 2
Why this answer is correct
The correct answer is C. (4). Fix (1), and the remaining two elements may be chosen or not chosen in \(2^2=4\) ways. With a fixed element, the rest are independent.
Step 3
Exam Tip
(1) को निश्चित रखकर शेष (2) तत्वों को चुनने या न चुनने के \(2^2=4\) तरीके हैं। निश्चित तत्व वाली गिनती में बाकी तत्व स्वतंत्र होते हैं।
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यदि (|U|=20), (|A|=12), (|B|=9) और \(|A\cap B|=5\), तो (|\(A\cup B\)'|) क्या होगा?
If (|U|=20), (|A|=12), (|B|=9) and \(|A\cap B|=5\), what is (|\(A\cup B\)'|)?
#sets
#universal-set
#venn-counting
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(|A\cup B|=12+9-5=16\). The complement has (20-16=4) elements.
Step 2
Why this answer is correct
The correct answer is B. (4). \(|A\cup B|=12+9-5=16\). The complement has (20-16=4) elements.
Step 3
Exam Tip
\(|A\cup B|=12+9-5=16\) है। पूरक की संख्या (20-16=4) होगी।
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किस स्थिति में \(\mathcal{P}(A)\subseteq \mathcal{P}(B)\) सत्य होगा?
In which condition is \(\mathcal{P}(A)\subseteq \mathcal{P}(B)\) true?
#sets
#power-set
#subset-relation
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A जब \(A\subseteq B\) / When \(A\subseteq B\)
B जब \(B\subseteq A\) / When \(B\subseteq A\)
C जब \(A\cap B=\emptyset\) / When \(A\cap B=\emptyset\)
D जब \(A\cup B=\emptyset\) / When \(A\cup B=\emptyset\)
Explanation opens after your attempt
Correct Answer
A. जब \(A\subseteq B\) / When \(A\subseteq B\)
Step 1
Concept
If \(A\subseteq B\), every subset of (A) is also a subset of (B). So the same inclusion holds for power sets.
Step 2
Why this answer is correct
The correct answer is A. जब \(A\subseteq B\) / When \(A\subseteq B\). If \(A\subseteq B\), every subset of (A) is also a subset of (B). So the same inclusion holds for power sets.
Step 3
Exam Tip
यदि \(A\subseteq B\), तो (A) का हर उपसमुच्चय (B) का भी उपसमुच्चय होगा। इसलिए घात समुच्चयों में भी यही समावेशन रहता है।
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यदि \(A=\{0,1,2,3\}\), तो (\mathcal{P}(A)) में वे कितने तत्व हैं जिनका योग (3) है?
If \(A=\{0,1,2,3\}\), how many elements of (\mathcal{P}(A)) have sum (3)?
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#power-set
#subset-sum
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The subsets with sum (3) are ({3},{0,3},{1,2},{0,1,2}). Hence there are (4).
Step 2
Why this answer is correct
The correct answer is C. (4). The subsets with sum (3) are ({3},{0,3},{1,2},{0,1,2}). Hence there are (4).
Step 3
Exam Tip
योग (3) वाले उपसमुच्चय ({3},{0,3},{1,2},{0,1,2}) हैं। इसलिए कुल (4) हैं।
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यदि \(U={x:x\in \mathbb{N}, x\le 12}\) और \(A={x:x\in U, x\) (3) का गुणज है(}), तो (A') में कितने तत्व होंगे?
If \(U={x:x\in \mathbb{N}, x\le 12}\) and \(A={x:x\in U, x\) is a multiple of (3)(}), how many elements are in (A')?
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#universal-set
#multiples
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A (4)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The multiples of (3) are (3,6,9,12), so (A) has (4) elements. Since (U) has (12) elements, (A') has (8) elements.
Step 2
Why this answer is correct
The correct answer is C. (8). The multiples of (3) are (3,6,9,12), so (A) has (4) elements. Since (U) has (12) elements, (A') has (8) elements.
Step 3
Exam Tip
(3) के गुणज (3,6,9,12) हैं, इसलिए (A) में (4) तत्व हैं। (U) में (12) तत्व हैं, अतः (A') में (8) तत्व होंगे।
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यदि \(A=\{m,n\}\), तो (\mathcal{P}(A)) का कौन सा उपसमुच्चय नहीं है?
If \(A=\{m,n\}\), which is not a subset of (\mathcal{P}(A))?
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#power-set
#second-level-subset
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A \({\emptyset,{m}}\)
B ({{m,n}})
C ({m,n})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
C. ({m,n})
Step 1
Concept
(\mathcal{P}(A)={\emptyset,{m},{n},{m,n}}). The elements (m,n) are not in it, so ({m,n}) is not its subset.
Step 2
Why this answer is correct
The correct answer is C. ({m,n}). (\mathcal{P}(A)={\emptyset,{m},{n},{m,n}}). The elements (m,n) are not in it, so ({m,n}) is not its subset.
Step 3
Exam Tip
(\mathcal{P}(A)={\emptyset,{m},{n},{m,n}}) है। ({m,n}) के तत्व (m,n) इसमें नहीं हैं, इसलिए यह इसका उपसमुच्चय नहीं है।
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यदि (A) में (n) तत्व हैं और (\mathcal{P}(A)) में (64) तत्व हैं, तो (A) के (3)-तत्वीय उपसमुच्चय कितने होंगे?
If (A) has (n) elements and (\mathcal{P}(A)) has (64) elements, how many (3)-element subsets does (A) have?
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#combination
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A (15)
B (18)
C (20)
D (30)
Explanation opens after your attempt
Step 1
Concept
Since \(2^n=64\), (n=6). The number of (3)-element subsets is \(\binom{6}{3}=20\).
Step 2
Why this answer is correct
The correct answer is C. (20). Since \(2^n=64\), (n=6). The number of (3)-element subsets is \(\binom{6}{3}=20\).
Step 3
Exam Tip
\(2^n=64\), इसलिए (n=6) है। (3)-तत्वीय उपसमुच्चय \(\binom{6}{3}=20\) होंगे।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,3,5,7\}\), तो ((A')') क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,3,5,7\}\), what is ((A')')?
#sets
#universal-set
#double-complement
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A ({2,4,6,8})
B ({1,3,5,7})
C \(\emptyset\)
D (U)
Explanation opens after your attempt
Correct Answer
B. ({1,3,5,7})
Step 1
Concept
The complement of a complement gives the original set. Hence ((A')'=A).
Step 2
Why this answer is correct
The correct answer is B. ({1,3,5,7}). The complement of a complement gives the original set. Hence ((A')'=A).
Step 3
Exam Tip
किसी समुच्चय का दो बार पूरक वही समुच्चय देता है। इसलिए ((A')'=A)।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (2) और (5) दोनों होंगे लेकिन (1) नहीं होगा?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) contain both (2) and (5) but not (1)?
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#restricted-subsets
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(2,5) are fixed and (1) is forbidden. The remaining (3,4) are free, so there are \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. (4). (2,5) are fixed and (1) is forbidden. The remaining (3,4) are free, so there are \(2^2=4\) subsets.
Step 3
Exam Tip
(2,5) निश्चित हैं और (1) निषिद्ध है। शेष (3,4) स्वतंत्र हैं, इसलिए \(2^2=4\) उपसमुच्चय होंगे।
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यदि \(U={x:x\in \mathbb{Z}, 0\le x\le 10}\), \(A=\{0,2,4,6,8,10\}\), तो (A') क्या है?
If \(U={x:x\in \mathbb{Z}, 0\le x\le 10}\), \(A=\{0,2,4,6,8,10\}\), what is (A')?
#sets
#universal-set
#even-odd
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A ({1,3,5,7,9})
B ({2,4,6,8,10})
C ({0,1,3,5,7,9})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,7,9})
Step 1
Concept
(A) contains the even numbers of (U). Its complement contains the odd numbers of (U).
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,7,9}). (A) contains the even numbers of (U). Its complement contains the odd numbers of (U).
Step 3
Exam Tip
(A) में (U) की सम संख्याएँ हैं। पूरक में (U) की विषम संख्याएँ रहेंगी।
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यदि (|U|=15), (|A'|=6), तो (|A|) क्या होगा?
If (|U|=15), (|A'|=6), what is (|A|)?
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#cardinality
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A (6)
B (9)
C (15)
D (21)
Explanation opens after your attempt
Step 1
Concept
A set and its complement together form (U). Hence (|A|=15-6=9).
Step 2
Why this answer is correct
The correct answer is B. (9). A set and its complement together form (U). Hence (|A|=15-6=9).
Step 3
Exam Tip
पूरक और समुच्चय मिलकर (U) बनाते हैं। इसलिए (|A|=15-6=9)।
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यदि \(A=\{a,b,c,d\}\), तो (\mathcal{P}(A)) में विषम संख्या वाले तत्वों के उपसमुच्चय कितने हैं?
If \(A=\{a,b,c,d\}\), how many elements of (\mathcal{P}(A)) are subsets with an odd number of elements?
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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
Odd-sized subsets are \(\binom{4}{1}+\binom{4}{3}=8\). For (n>0), half the subsets have odd size.
Step 2
Why this answer is correct
The correct answer is C. (8). Odd-sized subsets are \(\binom{4}{1}+\binom{4}{3}=8\). For (n>0), half the subsets have odd size.
Step 3
Exam Tip
विषम आकार वाले उपसमुच्चय \(\binom{4}{1}+\binom{4}{3}=8\) हैं। (n>0) में आधे उपसमुच्चय विषम आकार के होते हैं।
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यदि \(U=\{1,2,3,4,5,6,7\}\), \(A=\{1,2,4\}\), \(B=\{2,4,6\}\), तो (\(A\cap B\)') क्या है?
If \(U=\{1,2,3,4,5,6,7\}\), \(A=\{1,2,4\}\), \(B=\{2,4,6\}\), what is (\(A\cap B\)')?
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#universal-set
#intersection-complement
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A ({1,3,5,6,7})
B ({2,4})
C ({1,6})
D ({3,5,7})
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,6,7})
Step 1
Concept
\(A\cap B={2,4}\). The elements outside it in (U) are ({1,3,5,6,7}).
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,6,7}). \(A\cap B={2,4}\). The elements outside it in (U) are ({1,3,5,6,7}).
Step 3
Exam Tip
\(A\cap B={2,4}\) है। (U) में इसके बाहर ({1,3,5,6,7}) हैं।
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यदि (A) में (3) तत्व हैं, तो (\mathcal{P}(A)) में कुल कितने क्रमित युग्म ((X,Y)) होंगे जिनमें \(X\subseteq Y\subseteq A\)?
If (A) has (3) elements, how many ordered pairs ((X,Y)) are there such that \(X\subseteq Y\subseteq A\)?
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#power-set
#ordered-pairs
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A (8)
B (18)
C (27)
D (64)
Explanation opens after your attempt
Step 1
Concept
For each element there are three choices: in (X), only in (Y), or outside both. Thus the total is \(3^3=27\).
Step 2
Why this answer is correct
The correct answer is C. (27). For each element there are three choices: in (X), only in (Y), or outside both. Thus the total is \(3^3=27\).
Step 3
Exam Tip
हर तत्व के लिए तीन स्थितियाँ हैं: (X) में, केवल (Y) में, या दोनों से बाहर। इसलिए कुल \(3^3=27\) युग्म हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10,11,12\}\), \(A=\{2,4,6,8,10,12\}\), \(B=\{3,6,9,12\}\), तो (\(A\cup B\)') में कितने तत्व हैं?
If \(U=\{1,2,3,4,5,6,7,8,9,10,11,12\}\), \(A=\{2,4,6,8,10,12\}\), \(B=\{3,6,9,12\}\), how many elements are in (\(A\cup B\)')?
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#universal-set
#multiples
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={2,3,4,6,8,9,10,12}\). Its complement is ({1,5,7,11}), so it has (4) elements.
Step 2
Why this answer is correct
The correct answer is C. (4). \(A\cup B={2,3,4,6,8,9,10,12}\). Its complement is ({1,5,7,11}), so it has (4) elements.
Step 3
Exam Tip
\(A\cup B={2,3,4,6,8,9,10,12}\) है। पूरक ({1,5,7,11}) है, इसलिए (4) तत्व हैं।
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यदि \(A={\emptyset,0}\), तो (\mathcal{P}(A)) क्या है?
If \(A={\emptyset,0}\), what is (\mathcal{P}(A))?
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#power-set
#empty-set
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A \({\emptyset,{\emptyset},{0},{\emptyset,0}}\)
B \({\emptyset,0}\)
C \({{\emptyset},0}\)
D \({\emptyset,{0}}\)
Explanation opens after your attempt
Correct Answer
A. \({\emptyset,{\emptyset},{0},{\emptyset,0}}\)
Step 1
Concept
A two-element set has (4) subsets in its power set. Here \(\emptyset\) is also an element of (A).
Step 2
Why this answer is correct
The correct answer is A. \({\emptyset,{\emptyset},{0},{\emptyset,0}}\). A two-element set has (4) subsets in its power set. Here \(\emptyset\) is also an element of (A).
Step 3
Exam Tip
दो तत्वों वाले समुच्चय का घात समुच्चय (4) उपसमुच्चयों से बनता है। यहाँ \(\emptyset\) भी (A) का एक तत्व है।
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यदि (A) में (6) तत्व हैं, तो (\mathcal{P}(A)) के कितने तत्वों में ठीक (2) तत्व नहीं होंगे?
If (A) has (6) elements, how many elements of (\mathcal{P}(A)) do not have exactly (2) elements?
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#subset-counting
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A (15)
B (49)
C (57)
D (64)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^6=64\), and exactly two-element subsets are \(\binom{6}{2}=15\). So the answer is (64-15=49).
Step 2
Why this answer is correct
The correct answer is B. (49). Total subsets are \(2^6=64\), and exactly two-element subsets are \(\binom{6}{2}=15\). So the answer is (64-15=49).
Step 3
Exam Tip
कुल उपसमुच्चय \(2^6=64\) हैं और ठीक (2) तत्व वाले \(\binom{6}{2}=15\) हैं। इसलिए उत्तर (64-15=49) है।
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यदि (U) छात्रों का समुच्चय है, (A) गणित पढ़ने वालों का और (B) भौतिकी पढ़ने वालों का है, तो \(A'\cap B'\) क्या दर्शाता है?
If (U) is the set of students, (A) is students studying Mathematics and (B) is students studying Physics, what does \(A'\cap B'\) represent?
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#venn-interpretation
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A जो दोनों विषय पढ़ते हैं / Students studying both subjects
B जो कोई भी विषय नहीं पढ़ते / Students studying neither subject
C जो केवल गणित पढ़ते हैं / Students studying only Mathematics
D जो कम से कम एक विषय पढ़ते हैं / Students studying at least one subject
Explanation opens after your attempt
Correct Answer
B. जो कोई भी विषय नहीं पढ़ते / Students studying neither subject
Step 1
Concept
\(A'\cap B'\) contains students not in (A) and not in (B). It represents students studying neither Mathematics nor Physics.
Step 2
Why this answer is correct
The correct answer is B. जो कोई भी विषय नहीं पढ़ते / Students studying neither subject. \(A'\cap B'\) contains students not in (A) and not in (B). It represents students studying neither Mathematics nor Physics.
Step 3
Exam Tip
\(A'\cap B'\) में वे छात्र हैं जो (A) में भी नहीं और (B) में भी नहीं हैं। यह न गणित न भौतिकी पढ़ने वालों को दर्शाता है।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (1) नहीं होगा और (4) होगा?
If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) do not contain (1) and contain (4)?
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#power-set
#restricted-subsets
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
(4) is fixed and (1) is excluded. The remaining (2,3) give \(2^2=4\) choices.
Step 2
Why this answer is correct
The correct answer is C. (4). (4) is fixed and (1) is excluded. The remaining (2,3) give \(2^2=4\) choices.
Step 3
Exam Tip
(4) निश्चित है और (1) नहीं लेना है। शेष (2,3) के लिए \(2^2=4\) विकल्प हैं।
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यदि \(U=\{1,2,3,4,5\}\) और \(A=\emptyset\), तो (A') क्या होगा?
If \(U=\{1,2,3,4,5\}\) and \(A=\emptyset\), what is (A')?
#sets
#universal-set
#empty-set
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A \(\emptyset\)
B ({1,2,3,4,5})
C \({\emptyset}\)
D ({0})
Explanation opens after your attempt
Correct Answer
B. ({1,2,3,4,5})
Step 1
Concept
The complement of the empty set is the whole universal set. Hence (A'=U).
Step 2
Why this answer is correct
The correct answer is B. ({1,2,3,4,5}). The complement of the empty set is the whole universal set. Hence (A'=U).
Step 3
Exam Tip
रिक्त समुच्चय का पूरक पूरे सार्वत्रिक समुच्चय के बराबर होता है। इसलिए (A'=U)।
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यदि \(A=\{1,2,3\}\), तो निम्न में से कौन सा कथन गलत है?
If \(A=\{1,2,3\}\), which of the following statements is false?
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#power-set
#false-statement
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A \(\emptyset\in \mathcal{P}(A)\)
B \({1,2}\in \mathcal{P}(A)\)
C \(1\in \mathcal{P}(A)\)
D \(A\in \mathcal{P}(A)\)
Explanation opens after your attempt
Correct Answer
C. \(1\in \mathcal{P}(A)\)
Step 1
Concept
A power set contains subsets, not bare elements. (1) would need to be written as ({1}) to be an element of the power set.
Step 2
Why this answer is correct
The correct answer is C. \(1\in \mathcal{P}(A)\). A power set contains subsets, not bare elements. (1) would need to be written as ({1}) to be an element of the power set.
Step 3
Exam Tip
(\mathcal{P}(A)) में उपसमुच्चय होते हैं, अकेले तत्व नहीं। (1) को ({1}) लिखने पर वह घात समुच्चय का तत्व होगा।
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यदि \(U=\{a,b,c,d,e,f\}\), (A'={b,d,f}), तो (A) क्या है?
If \(U=\{a,b,c,d,e,f\}\), (A'={b,d,f}), what is (A)?
#sets
#universal-set
#complement-reversal
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A ({a,c,e})
B ({b,d,f})
C ({a,b,c,d,e,f})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({a,c,e})
Step 1
Concept
(A) is the set of elements left in (U) outside (A'). Therefore \(A=\{a,c,e\}\).
Step 2
Why this answer is correct
The correct answer is A. ({a,c,e}). (A) is the set of elements left in (U) outside (A'). Therefore \(A=\{a,c,e\}\).
Step 3
Exam Tip
(A), (U) में (A') के बाहर बचे तत्वों का समुच्चय है। इसलिए \(A=\{a,c,e\}\)।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) के कितने तत्वों का आकार सम है?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) have even cardinality?
#sets
#power-set
#even-cardinality
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A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
A set with (5) elements has (32) subsets. Half of them, (16), have even cardinality.
Step 2
Why this answer is correct
The correct answer is B. (16). A set with (5) elements has (32) subsets. Half of them, (16), have even cardinality.
Step 3
Exam Tip
(5) तत्वों वाले समुच्चय के कुल (32) उपसमुच्चय हैं। इनमें आधे यानी (16) सम आकार के होते हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,4,9\}\), तो (A') का सही वर्णन क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,4,9\}\), what is the correct description of (A')?
#sets
#universal-set
#perfect-squares
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A (U) में पूर्ण वर्गों का समुच्चय / Perfect squares in (U)
B (U) में अपूर्ण वर्ग संख्याएँ / Non-perfect-square numbers in (U)
C सभी प्राकृतिक संख्याएँ / All natural numbers
D सभी विषम संख्याएँ / All odd numbers
Explanation opens after your attempt
Correct Answer
B. (U) में अपूर्ण वर्ग संख्याएँ / Non-perfect-square numbers in (U)
Step 1
Concept
(A) contains the perfect squares (1,4,9) in (U). Its complement contains the remaining non-perfect-square numbers in (U).
Step 2
Why this answer is correct
The correct answer is B. (U) में अपूर्ण वर्ग संख्याएँ / Non-perfect-square numbers in (U). (A) contains the perfect squares (1,4,9) in (U). Its complement contains the remaining non-perfect-square numbers in (U).
Step 3
Exam Tip
(A) में (U) के पूर्ण वर्ग (1,4,9) हैं। पूरक में (U) की बाकी अपूर्ण वर्ग संख्याएँ होंगी।
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यदि \(A=\{a,b,c\}\), तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जो ({a,b}) के अधिसमुच्चय हैं?
If \(A=\{a,b,c\}\), how many elements of (\mathcal{P}(A)) are supersets of ({a,b})?
#sets
#power-set
#superset-counting
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
({a,b}) must be included and (c) is optional. Thus ({a,b}) and ({a,b,c}), total (2), work.
Step 2
Why this answer is correct
The correct answer is B. (2). ({a,b}) must be included and (c) is optional. Thus ({a,b}) and ({a,b,c}), total (2), work.
Step 3
Exam Tip
({a,b}) को रखना जरूरी है और (c) वैकल्पिक है। इसलिए ({a,b}) और ({a,b,c}), कुल (2) हैं।
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यदि (U) सभी अंकों का समुच्चय है और \(A=\{0,2,4,6,8\}\), तो (A') क्या है?
If (U) is the set of all digits and \(A=\{0,2,4,6,8\}\), what is (A')?
#sets
#universal-set
#digits
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A ({1,3,5,7,9})
B ({2,4,6,8})
C ({0,1,3,5,7,9})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,7,9})
Step 1
Concept
All digits are from (0) to (9). (A) is the set of even digits, so the complement is the odd digits.
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,7,9}). All digits are from (0) to (9). (A) is the set of even digits, so the complement is the odd digits.
Step 3
Exam Tip
सभी अंक (0) से (9) तक होते हैं। (A) सम अंकों का समुच्चय है, इसलिए पूरक विषम अंक हैं।
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यदि \(A=\{1,2,3,4\}\), तो (\mathcal{P}(A)) के कितने तत्वों में (1) और (2) में से ठीक एक होगा?
If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) contain exactly one of (1) and (2)?
#sets
#power-set
#exclusive-choice
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A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are (2) ways to choose exactly one of (1,2). The remaining (3,4) give \(2^2\) choices, so total (8).
Step 2
Why this answer is correct
The correct answer is C. (8). There are (2) ways to choose exactly one of (1,2). The remaining (3,4) give \(2^2\) choices, so total (8).
Step 3
Exam Tip
(1,2) में से ठीक एक चुनने के (2) तरीके हैं। शेष (3,4) के लिए \(2^2\) तरीके, इसलिए कुल (8) हैं।
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यदि \(U=\mathbb{N}\) और \(A={x:x\in \mathbb{N}, x\le 100}\), तो (A') क्या है?
If \(U=\mathbb{N}\) and \(A={x:x\in \mathbb{N}, x\le 100}\), what is (A')?
#sets
#universal-set
#infinite-set
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A \({x:x\in \mathbb{N}, x>100}\)
B \({x:x\in \mathbb{N}, x<100}\)
C ({100})
D \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. \({x:x\in \mathbb{N}, x>100}\)
Step 1
Concept
(A) contains natural numbers up to (100). In \(\mathbb{N}\), the complement is natural numbers greater than (100).
Step 2
Why this answer is correct
The correct answer is A. \({x:x\in \mathbb{N}, x>100}\). (A) contains natural numbers up to (100). In \(\mathbb{N}\), the complement is natural numbers greater than (100).
Step 3
Exam Tip
(A) में (100) तक की प्राकृतिक संख्याएँ हैं। \(\mathbb{N}\) में पूरक (100) से बड़ी प्राकृतिक संख्याएँ हैं।
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यदि \(A=\{1,2,3\}\), तो \(\mathcal{P}(A)\setminus{\emptyset,A}\) में कितने तत्व होंगे?
If \(A=\{1,2,3\}\), how many elements are in \(\mathcal{P}(A)\setminus{\emptyset,A}\)?
#sets
#power-set
#proper-nonempty-subsets
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^3=8\) elements. Removing \(\emptyset\) and (A) leaves (6) elements.
Step 2
Why this answer is correct
The correct answer is C. (6). (\mathcal{P}(A)) has \(2^3=8\) elements. Removing \(\emptyset\) and (A) leaves (6) elements.
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^3=8\) तत्व हैं। \(\emptyset\) और (A) हटाने पर (6) तत्व बचते हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,2,3,4\}\) और \(B=\{3,4,5,6\}\), तो \(A'\cup B'\) क्या है?
If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{1,2,3,4\}\) and \(B=\{3,4,5,6\}\), what is \(A'\cup B'\)?
#sets
#universal-set
#de-morgan
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A ({1,2,5,6,7,8})
B ({3,4})
C ({7,8})
D ({1,2,3,4,5,6})
Explanation opens after your attempt
Correct Answer
A. ({1,2,5,6,7,8})
Step 1
Concept
(A'={5,6,7,8}) and (B'={1,2,7,8}). Their union is ({1,2,5,6,7,8}).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,5,6,7,8}). (A'={5,6,7,8}) and (B'={1,2,7,8}). Their union is ({1,2,5,6,7,8}).
Step 3
Exam Tip
(A'={5,6,7,8}) और (B'={1,2,7,8}) है। उनका संघ ({1,2,5,6,7,8}) है।
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यदि \(A=\{x,y,z,w\}\), तो (\mathcal{P}(A)) में ऐसे कितने तत्व हैं जो ({x,z}) से असंयुक्त हैं?
If \(A=\{x,y,z,w\}\), how many elements of (\mathcal{P}(A)) are disjoint from ({x,z})?
#sets
#power-set
#disjoint-subsets
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
A disjoint subset cannot contain (x) or (z). Only (y,w) may be chosen, so there are \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). A disjoint subset cannot contain (x) or (z). Only (y,w) may be chosen, so there are \(2^2=4\).
Step 3
Exam Tip
असंयुक्त उपसमुच्चय में (x) और (z) नहीं आ सकते। केवल (y,w) चुने जा सकते हैं, इसलिए \(2^2=4\) हैं।
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यदि \(U=\{1,2,3,4,5,6,7,8,9,10\}\) और \(A={x:x\in U, x\) न तो (2) न (5) से विभाज्य है(}), तो (A') क्या है?
If \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A={x:x\in U, x\) is divisible by neither (2) nor (5)(}), what is (A')?
#sets
#universal-set
#divisibility
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A ({2,4,5,6,8,10})
B ({1,3,7,9})
C ({2,5,10})
D ({1,2,3,4,5,6,7,8,9,10})
Explanation opens after your attempt
Correct Answer
A. ({2,4,5,6,8,10})
Step 1
Concept
(A) contains (1,3,7,9). So the complement is the elements divisible by (2) or (5): ({2,4,5,6,8,10}).
Step 2
Why this answer is correct
The correct answer is A. ({2,4,5,6,8,10}). (A) contains (1,3,7,9). So the complement is the elements divisible by (2) or (5): ({2,4,5,6,8,10}).
Step 3
Exam Tip
(A) में (1,3,7,9) हैं। इसलिए पूरक (2) या (5) से विभाज्य तत्व ({2,4,5,6,8,10}) हैं।
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