यदि \(A=\{2,4,6\}\) और \(B={x:x\) संख्या 8 से छोटी धनात्मक सम संख्या है(}), तो सही कथन कौन सा है?
If \(A=\{2,4,6\}\) and \(B={x:x\) is a positive even number less than 8(}), which statement is correct?
#sets
#equal sets
#subset
#class 11
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both sets have exactly the elements 2, 4 and 6. Order or description does not change a set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both sets have exactly the elements 2, 4 and 6. Order or description does not change a set.
Step 3
Exam Tip
दोनों समुच्चयों के सभी अवयव 2, 4 और 6 हैं। क्रम या वर्णन बदलने से समुच्चय नहीं बदलता।
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यदि \(A={x:x\in \mathbb{Z}, x^2-4x+3=0}\) और \(B=\{1,3\}\), तो सही संबंध कौन सा है?
If \(A={x:x\in \mathbb{Z}, x^2-4x+3=0}\) and \(B=\{1,3\}\), which relation is correct?
#sets
#equal sets
#quadratic
#level 9
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The equation gives ((x-1)(x-3)=0), so \(A=\{1,3\}\). Equal sets have the same elements, order is not important.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The equation gives ((x-1)(x-3)=0), so \(A=\{1,3\}\). Equal sets have the same elements, order is not important.
Step 3
Exam Tip
समीकरण से ((x-1)(x-3)=0), इसलिए \(A=\{1,3\}\)। बराबर समुच्चय में अवयव वही होते हैं, क्रम जरूरी नहीं।
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समुच्चय \(A=\{1,1,2,3\}\) और \(B=\{3,2,1\}\) के लिए कौन सा निष्कर्ष सही है?
For the sets \(A=\{1,1,2,3\}\) and \(B=\{3,2,1\}\), which conclusion is correct?
#sets
#duplicate elements
#equal sets
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A \(A\ne B\) क्योंकि 1 दो बार लिखा है / \(A\ne B\) because 1 is written twice
B (A=B)
C \(A\subset B\) लेकिन \(B\not\subset A\) / \(A\subset B\) but \(B\not\subset A\)
D \(B\subset A\) लेकिन \(A\not\subset B\) / \(B\subset A\) but \(A\not\subset B\)
Explanation opens after your attempt
Step 1
Concept
Repeated elements are not counted in a set. In exams, first list the distinct elements.
Step 2
Why this answer is correct
The correct answer is B. (A=B). Repeated elements are not counted in a set. In exams, first list the distinct elements.
Step 3
Exam Tip
समुच्चय में किसी अवयव की पुनरावृत्ति नहीं गिनी जाती। परीक्षा में पहले वास्तविक अलग-अलग अवयव लिखें।
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यदि \(A=\{a,b,c\}\) और \(B=\{a,b,c,d\}\), तो (A) और (B) के बारे में सही कथन कौन सा है?
If \(A=\{a,b,c\}\) and \(B=\{a,b,c,d\}\), which statement about (A) and (B) is correct?
#proper subset
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#class 11
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D \(A\not\subset B\)
Explanation opens after your attempt
Correct Answer
B. \(A\subset B\)
Step 1
Concept
Every element of (A) is in (B), and (B) has one extra element. So (A) is a proper subset of (B).
Step 2
Why this answer is correct
The correct answer is B. \(A\subset B\). Every element of (A) is in (B), and (B) has one extra element. So (A) is a proper subset of (B).
Step 3
Exam Tip
(A) का हर अवयव (B) में है और (B) में एक अतिरिक्त अवयव है। इसलिए (A), (B) का उचित उपसमुच्चय है।
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यदि \(A=\{a,b,c,d,e\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (a) हो, (c) न हो और ठीक 3 अवयव हों?
If \(A=\{a,b,c,d,e\}\), how many subsets contain (a), do not contain (c), and have exactly 3 elements?
#subsets
#counting
#combination
#level 9
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A 2
B 3
C 4
D 6
Explanation opens after your attempt
Step 1
Concept
(a) is fixed and (c) is excluded, so choose 2 from (b,d,e). The number is \(\binom{3}{2}=3\).
Step 2
Why this answer is correct
The correct answer is B. 3. (a) is fixed and (c) is excluded, so choose 2 from (b,d,e). The number is \(\binom{3}{2}=3\).
Step 3
Exam Tip
(a) निश्चित है और (c) हटाया गया है, इसलिए (b,d,e) में से 2 चुनने हैं। संख्या \(\binom{3}{2}=3\) है।
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यदि ({x,5}={3,y}), तो निम्न में से कौन सा युग्म सही हो सकता है?
If ({x,5}={3,y}), which pair can be correct?
#equal sets
#variables
#reasoning
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A (x=3,y=5)
B (x=5,y=3)
C (x=3,y=3)
D (x=5,y=5)
Explanation opens after your attempt
Correct Answer
A. (x=3,y=5)
Step 1
Concept
Equal sets must contain exactly the same elements, so 3 and 5 must appear on both sides. Do not be confused by order.
Step 2
Why this answer is correct
The correct answer is A. (x=3,y=5). Equal sets must contain exactly the same elements, so 3 and 5 must appear on both sides. Do not be confused by order.
Step 3
Exam Tip
बराबर समुच्चयों में वही अवयव होने चाहिए, इसलिए 3 और 5 दोनों ओर होने चाहिए। ऐसे प्रश्नों में क्रम से भ्रमित न हों।
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यदि ({m,m+2}={4,6}), तो (m) का मान क्या है?
If ({m,m+2}={4,6}), what is the value of (m)?
#equal sets
#variable
#reasoning
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A 2
B 4
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
For equal sets, the two elements must be exactly 4 and 6. Substituting (m=4) gives ({m,m+2}={4,6}).
Step 2
Why this answer is correct
The correct answer is B. 4. For equal sets, the two elements must be exactly 4 and 6. Substituting (m=4) gives ({m,m+2}={4,6}).
Step 3
Exam Tip
बराबर समुच्चय के लिए दोनों अवयव 4 और 6 ही होने चाहिए। (m=4) रखने पर ({m,m+2}={4,6}) मिलता है।
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समुच्चय \(A={x:x^2=4,\ x\in \mathbb{Z}}\) और \(B=\{-2,2\}\) के लिए सही कथन चुनिए।
Choose the correct statement for \(A={x:x^2=4,\ x\in \mathbb{Z}}\) and \(B=\{-2,2\}\).
#equal sets
#roster form
#set builder
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A (A=B)
B \(A=\{2\}\) इसलिए \(A\ne B\) / \(A=\{2\}\) so \(A\ne B\)
C \(B\subset A\) लेकिन \(A\not\subset B\) / \(B\subset A\) but \(A\not\subset B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
In integers, the solutions of \(x^2=4\) are (-2) and 2. Both sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). In integers, the solutions of \(x^2=4\) are (-2) and 2. Both sets are equal.
Step 3
Exam Tip
पूर्णांकों में \(x^2=4\) के हल (-2) और 2 हैं। दोनों समुच्चय बराबर हैं।
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यदि \(A={\varnothing,{2},2}\), तो निम्न में से कौन सा कथन सही है?
If \(A={\varnothing,{2},2}\), which of the following statements is correct?
#element vs subset
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#notation
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A \({2}\in A\) और \({2}\subset A\) दोनों सत्य हैं / Both \({2}\in A\) and \({2}\subset A\) are true
B \(2\notin A\)
C \(\varnothing\not\subset A\)
D \(A=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \({2}\in A\) और \({2}\subset A\) दोनों सत्य हैं / Both \({2}\in A\) and \({2}\subset A\) are true
Step 1
Concept
({2}) is an element of (A), and 2 is also in (A), so \({2}\subset A\) is also true. Keep the difference between \(\in\) and \(\subset\) clear.
Step 2
Why this answer is correct
The correct answer is A. \({2}\in A\) और \({2}\subset A\) दोनों सत्य हैं / Both \({2}\in A\) and \({2}\subset A\) are true. ({2}) is an element of (A), and 2 is also in (A), so \({2}\subset A\) is also true. Keep the difference between \(\in\) and \(\subset\) clear.
Step 3
Exam Tip
({2}) (A) का अवयव है और 2 भी (A) में है, इसलिए \({2}\subset A\) भी सत्य है। \(\in\) और \(\subset\) का अंतर ध्यान रखें।
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यदि \(A={x:x\) 12 का धनात्मक गुणनखंड है(}) और \(B=\{1,2,3,4,6,12\}\), तो क्या सत्य है?
If \(A={x:x\) is a positive factor of 12(}) and \(B=\{1,2,3,4,6,12\}\), what is true?
#factors
#equal sets
#set builder
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D (A) अनंत है / (A) is infinite
Explanation opens after your attempt
Step 1
Concept
These six numbers are exactly the positive factors of 12. Set-builder form and roster form give the same set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). These six numbers are exactly the positive factors of 12. Set-builder form and roster form give the same set.
Step 3
Exam Tip
12 के सभी धनात्मक गुणनखंड यही छह हैं। वर्णनात्मक रूप और सूची रूप एक ही समुच्चय दे रहे हैं।
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यदि \(A\subseteq B\), (A) और (B) दोनों सीमित समुच्चय हैं और (n(A)=n(B)=7), तो निष्कर्ष क्या होगा?
If \(A\subseteq B\), (A) and (B) are finite sets, and (n(A)=n(B)=7), what will be the conclusion?
#finite sets
#subset
#equal sets
#proof
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
For finite sets, if one is a subset of the other and both have the same number of elements, they are equal. This is a useful rule for proving equality.
Step 2
Why this answer is correct
The correct answer is A. (A=B). For finite sets, if one is a subset of the other and both have the same number of elements, they are equal. This is a useful rule for proving equality.
Step 3
Exam Tip
सीमित समुच्चयों में यदि एक दूसरे का उपसमुच्चय हो और दोनों में समान संख्या में अवयव हों, तो वे बराबर होते हैं। यह बराबरी सिद्ध करने का उपयोगी नियम है।
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यदि \(A=\{1,2,{3}\}\), तो कौन सा कथन सही है?
If \(A=\{1,2,{3}\}\), which statement is correct?
#element
#subset
#common mistake
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A \(3\in A\)
B \({3}\in A\)
C ({1,2,3}=A)
D \({3}\subset A\)
Explanation opens after your attempt
Correct Answer
B. \({3}\in A\)
Step 1
Concept
Here ({3}) itself is an element of (A), not 3. Keep element and subset ideas separate.
Step 2
Why this answer is correct
The correct answer is B. \({3}\in A\). Here ({3}) itself is an element of (A), not 3. Keep element and subset ideas separate.
Step 3
Exam Tip
यहां ({3}) स्वयं (A) का अवयव है, 3 नहीं। अवयव और उपसमुच्चय में अंतर ध्यान रखें।
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यदि \(A=\{1,2,3\}\), तो निम्न में से कौन सा (A) का उपसमुच्चय नहीं है?
If \(A=\{1,2,3\}\), which of the following is not a subset of (A)?
#subset
#not subset
#class 11
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A \(\varnothing\)
B ({1,3})
C ({2,4})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
C. ({2,4})
Step 1
Concept
4 is not an element of (A), so ({2,4}) is not a subset. Look for any outside element in the option.
Step 2
Why this answer is correct
The correct answer is C. ({2,4}). 4 is not an element of (A), so ({2,4}) is not a subset. Look for any outside element in the option.
Step 3
Exam Tip
4, (A) का अवयव नहीं है, इसलिए ({2,4}) उपसमुच्चय नहीं है। किसी भी विकल्प में बाहरी अवयव खोजें।
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किसी 5 अवयव वाले समुच्चय के कुल उपसमुच्चयों की संख्या क्या होगी?
What is the total number of subsets of a set having 5 elements?
#number of subsets
#power set
#hard
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A 10
B 25
C 32
D 120
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets. Here \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is C. 32. A set with (n) elements has \(2^n\) subsets. Here \(2^5=32\).
Step 3
Exam Tip
(n) अवयवों वाले समुच्चय के उपसमुच्चय \(2^n\) होते हैं। यहां \(2^5=32\) है।
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यदि (A) के कुल उपसमुच्चय 64 हैं, तो (A) में कितने अवयव हैं?
If (A) has 64 subsets, how many elements does (A) have?
#power set
#subset count
#exam
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A 5
B 6
C 7
D 8
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^n=64\), so (n=6). Recognising powers is the quick method.
Step 2
Why this answer is correct
The correct answer is B. 6. Total subsets are \(2^n=64\), so (n=6). Recognising powers is the quick method.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^n=64\) हैं, इसलिए (n=6)। घातों को पहचानना तेज तरीका है।
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4 अवयव वाले समुच्चय के उचित उपसमुच्चयों की संख्या क्या है?
What is the number of proper subsets of a 4-element set?
#proper subset
#subset count
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A 15
B 16
C 8
D 14
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\), and the set itself is not a proper subset. So the number is 15.
Step 2
Why this answer is correct
The correct answer is A. 15. Total subsets are \(2^4=16\), and the set itself is not a proper subset. So the number is 15.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं और समुच्चय स्वयं उचित उपसमुच्चय नहीं है। इसलिए संख्या 15 है।
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यदि \(A=\varnothing\), तो (A) के उपसमुच्चयों की संख्या कितनी है?
If \(A=\varnothing\), how many subsets does (A) have?
#empty set
#subset count
#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 exactly one subset, the empty set itself. Remember \(2^0=1\).
Step 2
Why this answer is correct
The correct answer is B. 1. The empty set has exactly one subset, the empty set itself. Remember \(2^0=1\).
Step 3
Exam Tip
रिक्त समुच्चय का एक ही उपसमुच्चय है, वह स्वयं रिक्त समुच्चय है। \(2^0=1\) याद रखें।
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यदि \(A=\{p,q,r,s\}\), तो (A) के 2 अवयव वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{p,q,r,s\}\), how many 2-element subsets of (A) are there?
#subset count
#combination
#hard
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A 4
B 5
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose 2 elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is C. 6. The number of ways to choose 2 elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 3
Exam Tip
2 अवयव चुनने के तरीके \(\binom{4}{2}=6\) हैं। उपसमुच्चयों में क्रम नहीं गिना जाता।
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यदि \(A=\{1,2,3,4,5\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 अवश्य हो?
If \(A=\{1,2,3,4,5\}\), how many subsets must contain 1?
#subsets
#contains element
#counting
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A 8
B 16
C 24
D 32
Explanation opens after your attempt
Step 1
Concept
Fix 1, and each of the remaining 4 elements may be chosen or not. So there are \(2^4=16\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 16. Fix 1, and each of the remaining 4 elements may be chosen or not. So there are \(2^4=16\) subsets.
Step 3
Exam Tip
1 को निश्चित रखें और शेष 4 अवयवों को चुनना या न चुनना स्वतंत्र है। इसलिए \(2^4=16\) उपसमुच्चय मिलते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 2 नहीं हो?
If \(A=\{1,2,3,4,5,6\}\), how many subsets do not contain 2?
#subset counting
#excluded element
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A 16
B 32
C 48
D 64
Explanation opens after your attempt
Step 1
Concept
After excluding 2, 5 elements remain. Their subsets are \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. 32. After excluding 2, 5 elements remain. Their subsets are \(2^5=32\).
Step 3
Exam Tip
2 को हटाने के बाद 5 अवयव बचते हैं। इनके सभी उपसमुच्चय \(2^5=32\) हैं।
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यदि \(A=\{a,b,c,d,e\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (a) और (b) दोनों हों?
If \(A=\{a,b,c,d,e\}\), how many subsets contain both (a) and (b)?
#subset counting
#fixed elements
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A 4
B 8
C 16
D 32
Explanation opens after your attempt
Step 1
Concept
(a) and (b) are fixed, while the other 3 elements are optional. Hence \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. (a) and (b) are fixed, while the other 3 elements are optional. Hence \(2^3=8\) subsets.
Step 3
Exam Tip
(a) और (b) निश्चित हैं, बाकी 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A=\{1,2,3,4\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक 3 अवयव हों?
If \(A=\{1,2,3,4\}\), how many subsets have exactly 3 elements?
#exact size subset
#combination
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A 3
B 4
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly 3 elements is \(\binom{4}{3}=4\). Changing order does not create a new subset.
Step 2
Why this answer is correct
The correct answer is B. 4. The number of ways to choose exactly 3 elements is \(\binom{4}{3}=4\). Changing order does not create a new subset.
Step 3
Exam Tip
ठीक 3 अवयव चुनने के तरीके \(\binom{4}{3}=4\) हैं। क्रम बदलने से नया उपसमुच्चय नहीं बनता।
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यदि \(A={x:x\) 10 से छोटी अभाज्य संख्या है(}) और \(B=\{2,3,5,7\}\), तो कौन सा कथन सही है?
If \(A={x:x\) is a prime number less than 10(}) and \(B=\{2,3,5,7\}\), which statement is correct?
#prime numbers
#equal sets
#common mistake
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(B\subset A\) और \(A\ne B\) / \(B\subset A\) and \(A\ne B\)
D \(A=\{1,2,3,5,7\}\)
Explanation opens after your attempt
Step 1
Concept
The prime numbers less than 10 are 2, 3, 5 and 7. The number 1 is not prime.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The prime numbers less than 10 are 2, 3, 5 and 7. The number 1 is not prime.
Step 3
Exam Tip
10 से छोटी अभाज्य संख्याएं 2, 3, 5 और 7 हैं। 1 अभाज्य नहीं है।
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यदि \(A={x:x^2-5x+6=0}\) और \(B=\{2,3\}\), तो सही कथन कौन सा है?
If \(A={x:x^2-5x+6=0}\) and \(B=\{2,3\}\), which statement is correct?
#quadratic
#equal sets
#set builder
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A (A=B)
B \(A=\{6\}\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The equation gives ((x-2)(x-3)=0), so the solutions are 2 and 3. The solution set equals (B).
Step 2
Why this answer is correct
The correct answer is A. (A=B). The equation gives ((x-2)(x-3)=0), so the solutions are 2 and 3. The solution set equals (B).
Step 3
Exam Tip
समीकरण ((x-2)(x-3)=0) देता है, इसलिए हल 2 और 3 हैं। हलों का समुच्चय (B) के बराबर है।
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यदि \(A=\{0,1\}\) और \(B={x:x^2=x}\), तो सही संबंध क्या है?
If \(A=\{0,1\}\) and \(B={x:x^2=x}\), what is the correct relation?
#equal sets
#solution set
#algebra
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B=\{1\}\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
From \(x^2=x\), (x(x-1)=0), so (x=0) or (x=1). Both sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). From \(x^2=x\), (x(x-1)=0), so (x=0) or (x=1). Both sets are equal.
Step 3
Exam Tip
\(x^2=x\) से (x(x-1)=0), इसलिए (x=0) या (x=1)। दोनों समुच्चय समान हैं।
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यदि \(A=\{1,2,3\}\), तो (P(A)) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), how many elements will (P(A)) have?
#power set
#subset
#count
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A 3
B 6
C 8
D 9
Explanation opens after your attempt
Step 1
Concept
(P(A)) contains all subsets of (A). For 3 elements, the number is \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. 8. (P(A)) contains all subsets of (A). For 3 elements, the number is \(2^3=8\).
Step 3
Exam Tip
(P(A)) में (A) के सभी उपसमुच्चय होते हैं। 3 अवयवों के लिए संख्या \(2^3=8\) है।
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यदि \(A\subseteq B\), तो \(A\cup B\) किसके बराबर होगा?
If \(A\subseteq B\), then \(A\cup B\) is equal to what?
#subset
#union
#concept
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A (A)
B (B)
C \(\varnothing\)
D \(A\cap B\) हमेशा रिक्त / \(A\cap B\) always empty
Explanation opens after your attempt
Step 1
Concept
When every element of (A) is already in (B), union adds nothing new. Hence \(A\cup B=B\).
Step 2
Why this answer is correct
The correct answer is B. (B). When every element of (A) is already in (B), union adds nothing new. Hence \(A\cup B=B\).
Step 3
Exam Tip
जब (A) का हर अवयव (B) में है, तो मिलाने पर नया अवयव नहीं आता। इसलिए \(A\cup B=B\)।
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यदि \(A\subseteq B\), तो \(A\cap B\) किसके बराबर होगा?
If \(A\subseteq B\), then \(A\cap B\) is equal to what?
#subset
#intersection
#concept
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A (A)
B (B)
C \(\varnothing\)
D \(A\cup B\)
Explanation opens after your attempt
Step 1
Concept
Every element of (A) is in (B), so the common elements are exactly (A). This is an important result of subset relation.
Step 2
Why this answer is correct
The correct answer is A. (A). Every element of (A) is in (B), so the common elements are exactly (A). This is an important result of subset relation.
Step 3
Exam Tip
(A) का हर अवयव (B) में है, इसलिए साझा अवयव पूरे (A) ही हैं। यह उपसमुच्चय का महत्वपूर्ण परिणाम है।
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यदि \(A=\{1,2\}\), तो (P(A)) कौन सा है?
If \(A=\{1,2\}\), which is (P(A))?
#power set
#subset notation
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A \({\varnothing,1,2}\)
B \({\varnothing,{1},{2},{1,2}}\)
C ({{1},{2}})
D ({1,2,{1,2}})
Explanation opens after your attempt
Correct Answer
B. \({\varnothing,{1},{2},{1,2}}\)
Step 1
Concept
In a power set, all subsets are written as elements. Thus single elements appear as ({1}) and ({2}).
Step 2
Why this answer is correct
The correct answer is B. \({\varnothing,{1},{2},{1,2}}\). In a power set, all subsets are written as elements. Thus single elements appear as ({1}) and ({2}).
Step 3
Exam Tip
पावर सेट में सभी उपसमुच्चय अवयव के रूप में लिखे जाते हैं। इसलिए एकल अवयव भी ({1}) और ({2}) रूप में होंगे।
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यदि \(A=\varnothing\) और \(B=\{0\}\), तो सही कथन कौन सा है?
If \(A=\varnothing\) and \(B=\{0\}\), which statement is correct?
#empty set
#proper subset
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A (A=B)
B \(A\subset B\)
C \(0\in A\)
D \(B\subset A\)
Explanation opens after your attempt
Correct Answer
B. \(A\subset B\)
Step 1
Concept
The empty set is a subset of every set, and (B) is not empty. Hence (A) is a proper subset.
Step 2
Why this answer is correct
The correct answer is B. \(A\subset B\). The empty set is a subset of every set, and (B) is not empty. Hence (A) is a proper subset.
Step 3
Exam Tip
रिक्त समुच्चय हर समुच्चय का उपसमुच्चय है और (B) रिक्त नहीं है। इसलिए (A) उचित उपसमुच्चय है।
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यदि \(A={x:x\) 6 का गुणज है और (x<20)(}) तथा \(B=\{6,12,18\}\), तो कौन सा संबंध सही है?
If \(A={x:x\) is a multiple of 6 and (x<20)(}) and \(B=\{6,12,18\}\), which relation is correct?
#multiples
#equal sets
#set builder
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D (A) अनंत है / (A) is infinite
Explanation opens after your attempt
Step 1
Concept
The positive multiples of 6 less than 20 are 6, 12 and 18. So the two sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive multiples of 6 less than 20 are 6, 12 and 18. So the two sets are equal.
Step 3
Exam Tip
20 से छोटे धनात्मक 6 के गुणज 6, 12 और 18 हैं। इसलिए दोनों समुच्चय समान हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4\}\), तो कौन सा कथन गलत है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4\}\), which statement is false?
#proper subset
#false statement
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A \(B\subset A\)
B \(B\subseteq A\)
C \(A\subseteq A\)
D \(A\subset B\)
Explanation opens after your attempt
Correct Answer
D. \(A\subset B\)
Step 1
Concept
(A) is a subset of itself, but it is not a proper subset of itself. Also \(A\subset B\) is false here.
Step 2
Why this answer is correct
The correct answer is D. \(A\subset B\). (A) is a subset of itself, but it is not a proper subset of itself. Also \(A\subset B\) is false here.
Step 3
Exam Tip
(A) स्वयं (A) का उपसमुच्चय है, लेकिन अपना उचित उपसमुच्चय नहीं है। इसलिए \(A\subset B\) भी असत्य है।
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यदि \(A=\{1,3,5,7\}\) और \(B={x:x\) 8 से छोटी धनात्मक विषम संख्या है(}), तो (A) और (B) कैसे संबंधित हैं?
If \(A=\{1,3,5,7\}\) and \(B={x:x\) is a positive odd number less than 8(}), how are (A) and (B) related?
#odd numbers
#equal sets
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The positive odd numbers less than 8 are 1, 3, 5 and 7. The list and rule give the same set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive odd numbers less than 8 are 1, 3, 5 and 7. The list and rule give the same set.
Step 3
Exam Tip
8 से छोटी धनात्मक विषम संख्याएं 1, 3, 5 और 7 हैं। सूची और नियम दोनों समान समुच्चय देते हैं।
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यदि \(A=\{2,3,5,7,11\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 2 हो लेकिन 11 न हो?
If \(A=\{2,3,5,7,11\}\), how many subsets contain 2 but not 11?
#subset counting
#conditions
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A 4
B 8
C 16
D 32
Explanation opens after your attempt
Step 1
Concept
2 is fixed and 11 is excluded, while the remaining 3 elements are optional. Hence \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. 2 is fixed and 11 is excluded, while the remaining 3 elements are optional. Hence \(2^3=8\) subsets.
Step 3
Exam Tip
2 निश्चित है और 11 निषिद्ध है, शेष 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय हैं।
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यदि \(A=\{a,b,c,d\}\), तो (A) के उन उपसमुच्चयों की संख्या कितनी है जिनमें कम से कम एक अवयव हो?
If \(A=\{a,b,c,d\}\), how many subsets of (A) have at least one element?
#nonempty subsets
#counting
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A 15
B 16
C 8
D 4
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\). For at least one element, remove the empty set, giving 15.
Step 2
Why this answer is correct
The correct answer is A. 15. Total subsets are \(2^4=16\). For at least one element, remove the empty set, giving 15.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं। कम से कम एक अवयव के लिए रिक्त समुच्चय हटाएं, इसलिए 15।
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यदि \(A=\{1,2,3,4,5\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक 2 अवयव हों और 5 न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets have exactly 2 elements and do not contain 5?
#exact subset
#counting
#combination
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A 4
B 6
C 8
D 10
Explanation opens after your attempt
Step 1
Concept
Exclude 5 and choose 2 elements from ({1,2,3,4}). The number is \(\binom{4}{2}=6\).
Step 2
Why this answer is correct
The correct answer is B. 6. Exclude 5 and choose 2 elements from ({1,2,3,4}). The number is \(\binom{4}{2}=6\).
Step 3
Exam Tip
5 को छोड़कर ({1,2,3,4}) से 2 अवयव चुनने हैं। संख्या \(\binom{4}{2}=6\) है।
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यदि \(A={x:x\in \mathbb{N}, 2x+1<10}\), तो (A) किसके बराबर है?
If \(A={x:x\in \mathbb{N}, 2x+1<10}\), what is (A) equal to?
#set builder
#inequality
#equal sets
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A ({1,2,3,4})
B ({0,1,2,3,4})
C ({1,2,3,4,5})
D ({2,3,4})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3,4})
Step 1
Concept
From (2x+1<10), we get (x<4.5). The natural numbers are (1,2,3,4).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,4}). From (2x+1<10), we get (x<4.5). The natural numbers are (1,2,3,4).
Step 3
Exam Tip
(2x+1<10) से (x<4.5) मिलता है। प्राकृतिक संख्याएं (1,2,3,4) हैं।
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यदि \(A=\{1,2,{1,2}\}\), तो निम्न में से कौन सा कथन सही है?
If \(A=\{1,2,{1,2}\}\), which of the following is correct?
#element
#subset
#advanced notation
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A \({1,2}\subset A\) और \({1,2}\in A\) दोनों सत्य हैं / Both \({1,2}\subset A\) and \({1,2}\in A\) are true
B \({1,2}\subset A\) सत्य है लेकिन \({1,2}\in A\) असत्य है / \({1,2}\subset A\) is true but \({1,2}\in A\) is false
C \({1,2}\in A\) सत्य है लेकिन \({1,2}\subset A\) असत्य है / \({1,2}\in A\) is true but \({1,2}\subset A\) is false
D दोनों असत्य हैं / Both are false
Explanation opens after your attempt
Correct Answer
A. \({1,2}\subset A\) और \({1,2}\in A\) दोनों सत्य हैं / Both \({1,2}\subset A\) and \({1,2}\in A\) are true
Step 1
Concept
Since (1) and (2) are in (A), \({1,2}\subset A\), and ({1,2}) is also an element. This is a good mixed notation example.
Step 2
Why this answer is correct
The correct answer is A. \({1,2}\subset A\) और \({1,2}\in A\) दोनों सत्य हैं / Both \({1,2}\subset A\) and \({1,2}\in A\) are true. Since (1) and (2) are in (A), \({1,2}\subset A\), and ({1,2}) is also an element. This is a good mixed notation example.
Step 3
Exam Tip
(1) और (2) (A) में हैं, इसलिए \({1,2}\subset A\), और ({1,2}) भी एक अवयव है। यह मिश्रित notation का अच्छा उदाहरण है।
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यदि \(A=\{1,2,3\}\), तो निम्न में से कौन सा (P(A)) का अवयव है?
If \(A=\{1,2,3\}\), which of the following is an element of (P(A))?
#power set
#element
#subset
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A 1
B ({1,2})
C ({4})
D ({1,4})
Explanation opens after your attempt
Correct Answer
B. ({1,2})
Step 1
Concept
Elements of (P(A)) are subsets of (A). ({1,2}) is a subset of (A).
Step 2
Why this answer is correct
The correct answer is B. ({1,2}). Elements of (P(A)) are subsets of (A). ({1,2}) is a subset of (A).
Step 3
Exam Tip
(P(A)) के अवयव (A) के उपसमुच्चय होते हैं। ({1,2}), (A) का उपसमुच्चय है।
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यदि \(A=\{1,2,3,4,5,6\}\), तो (A) के सम संख्या वाले सभी अवयवों का समुच्चय (B) क्या है?
If \(A=\{1,2,3,4,5,6\}\), what is the set (B) of all even elements of (A)?
#subset
#even numbers
#selection
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A ({1,3,5})
B ({2,4,6})
C ({2,4})
D ({1,2,3,4,5,6})
Explanation opens after your attempt
Correct Answer
B. ({2,4,6})
Step 1
Concept
The even elements in (A) are 2, 4 and 6. Thus \(B\subset A\) and \(B=\{2,4,6\}\).
Step 2
Why this answer is correct
The correct answer is B. ({2,4,6}). The even elements in (A) are 2, 4 and 6. Thus \(B\subset A\) and \(B=\{2,4,6\}\).
Step 3
Exam Tip
(A) में सम अवयव 2, 4 और 6 हैं। इसलिए \(B\subset A\) और \(B=\{2,4,6\}\)।
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यदि \(A={x:x\) 15 का धनात्मक भाजक है(}) और \(B=\{1,3,5,15\}\), तो कौन सा कथन सही है?
If \(A={x:x\) is a positive divisor of 15(}) and \(B=\{1,3,5,15\}\), which statement is correct?
#divisors
#equal sets
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{3,5\}\)
Explanation opens after your attempt
Step 1
Concept
The positive divisors of 15 are 1, 3, 5 and 15. Therefore the sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive divisors of 15 are 1, 3, 5 and 15. Therefore the sets are equal.
Step 3
Exam Tip
15 के धनात्मक भाजक 1, 3, 5 और 15 हैं। इसलिए दोनों समुच्चय बराबर हैं।
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यदि \(A=\{a,b,c\}\) और \(B=\{b,c,d\}\), तो कौन सा कथन सही है?
If \(A=\{a,b,c\}\) and \(B=\{b,c,d\}\), which statement is correct?
#not subset
#equal sets
#comparison
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D न तो \(A\subseteq B\) और न ही \(B\subseteq A\) / Neither \(A\subseteq B\) nor \(B\subseteq A\)
Explanation opens after your attempt
Correct Answer
D. न तो \(A\subseteq B\) और न ही \(B\subseteq A\) / Neither \(A\subseteq B\) nor \(B\subseteq A\)
Step 1
Concept
(a) is not in (B), and (d) is not in (A). Hence neither set is a subset of the other.
Step 2
Why this answer is correct
The correct answer is D. न तो \(A\subseteq B\) और न ही \(B\subseteq A\) / Neither \(A\subseteq B\) nor \(B\subseteq A\). (a) is not in (B), and (d) is not in (A). Hence neither set is a subset of the other.
Step 3
Exam Tip
(a), (B) में नहीं है और (d), (A) में नहीं है। इसलिए कोई भी दूसरे का उपसमुच्चय नहीं है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,3\}\), तो (A-B) के आधार पर कौन सा कथन सही है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,3\}\), which statement based on (A-B) is correct?
#set difference
#subset
#application
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A (A-B={1,4}) और यह (A) का उपसमुच्चय है / (A-B={1,4}) and it is a subset of (A)
B (A-B={2,3}) और यह (B) का उपसमुच्चय है / (A-B={2,3}) and it is a subset of (B)
C (A-B=A)
D \(A-B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. (A-B={1,4}) और यह (A) का उपसमुच्चय है / (A-B={1,4}) and it is a subset of (A)
Step 1
Concept
(A-B) contains elements of (A) that are not in (B). Here they are 1 and 4.
Step 2
Why this answer is correct
The correct answer is A. (A-B={1,4}) और यह (A) का उपसमुच्चय है / (A-B={1,4}) and it is a subset of (A). (A-B) contains elements of (A) that are not in (B). Here they are 1 and 4.
Step 3
Exam Tip
(A-B) में (A) के वे अवयव हैं जो (B) में नहीं हैं। यहां वे 1 और 4 हैं।
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कथन पर विचार करें: हर समुच्चय स्वयं का उपसमुच्चय है। इसका सही मूल्य क्या है?
Consider the statement: Every set is a subset of itself. What is its truth value?
#subset property
#reflexive
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A सत्य / True
B असत्य / False
C केवल रिक्त समुच्चय के लिए सत्य / True only for empty set
D केवल सीमित समुच्चय के लिए सत्य / True only for finite sets
Explanation opens after your attempt
Correct Answer
A. सत्य / True
Step 1
Concept
Every element of a set is in the same set, so \(A\subseteq A\) is true. Remember it like a reflexive property.
Step 2
Why this answer is correct
The correct answer is A. सत्य / True. Every element of a set is in the same set, so \(A\subseteq A\) is true. Remember it like a reflexive property.
Step 3
Exam Tip
हर अवयव अपने ही समुच्चय में होता है, इसलिए \(A\subseteq A\) सत्य है। इसे reflexive property की तरह याद रखें।
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कथन पर विचार करें: यदि \(A\subset B\), तो \(A\ne B\)। यह कथन कैसा है?
Consider the statement: If \(A\subset B\), then \(A\ne B\). What is the nature of this statement?
#proper subset
#notation
#true false
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A सत्य / True
B असत्य / False
C केवल अनंत समुच्चयों के लिए सत्य / True only for infinite sets
D केवल रिक्त समुच्चय के लिए असत्य / False only for empty set
Explanation opens after your attempt
Correct Answer
A. सत्य / True
Step 1
Concept
\(A\subset B\) denotes a proper subset, so equality is not possible. Distinguish \(\subset\) from \(\subseteq\).
Step 2
Why this answer is correct
The correct answer is A. सत्य / True. \(A\subset B\) denotes a proper subset, so equality is not possible. Distinguish \(\subset\) from \(\subseteq\).
Step 3
Exam Tip
\(A\subset B\) उचित उपसमुच्चय को दर्शाता है, इसलिए बराबरी संभव नहीं। प्रतीक \(\subset\) और \(\subseteq\) में अंतर रखें।
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यदि \(A=\{1,2,3,4\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 हो या 2 हो?
If \(A=\{1,2,3,4\}\), how many subsets contain 1 or 2?
#subset counting
#or condition
#hard
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A 8
B 12
C 14
D 16
Explanation opens after your attempt
Step 1
Concept
There are 16 total subsets, and those containing neither 1 nor 2 are the \(2^2=4\) subsets of ({3,4}). Hence (16-4=12).
Step 2
Why this answer is correct
The correct answer is B. 12. There are 16 total subsets, and those containing neither 1 nor 2 are the \(2^2=4\) subsets of ({3,4}). Hence (16-4=12).
Step 3
Exam Tip
कुल 16 उपसमुच्चय हैं और जिनमें न 1 है न 2, वे ({3,4}) के \(2^2=4\) उपसमुच्चय हैं। इसलिए (16-4=12)।
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यदि \(A=\{1,2,3,4,5\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 हो और 2 न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets contain 1 and do not contain 2?
#subset counting
#and condition
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A 4
B 8
C 16
D 24
Explanation opens after your attempt
Step 1
Concept
1 is fixed and 2 is excluded, while the remaining 3 elements are optional. Thus \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. 1 is fixed and 2 is excluded, while the remaining 3 elements are optional. Thus \(2^3=8\) subsets.
Step 3
Exam Tip
1 निश्चित है और 2 नहीं लेना है, शेष 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A={x:x\in \mathbb{N}, x\le 4}\) और \(B={x:x\) 5 से छोटी प्राकृतिक संख्या है(}), तो कौन सा कथन सही है?
If \(A={x:x\in \mathbb{N}, x\le 4}\) and \(B={x:x\) is a natural number less than 5(}), which statement is correct?
#equal sets
#natural numbers
#inequality
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both descriptions give ({1,2,3,4}). Do not be confused by different forms of inequalities.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both descriptions give ({1,2,3,4}). Do not be confused by different forms of inequalities.
Step 3
Exam Tip
दोनों वर्णन ({1,2,3,4}) देते हैं। असमानता के अलग रूपों से भ्रमित न हों।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\), तो (B) के ऐसे उपसमुच्चयों की संख्या कितनी है जो (A) को अवश्य समाहित करते हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many subsets of (B) must contain (A)?
#subset containing set
#counting
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A 2
B 4
C 8
D 16
Explanation opens after your attempt
Step 1
Concept
1, 2 and 3 are fixed, while 4 and 5 are optional. Hence \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 4. 1, 2 and 3 are fixed, while 4 and 5 are optional. Hence \(2^2=4\) subsets.
Step 3
Exam Tip
1, 2 और 3 निश्चित हैं, जबकि 4 और 5 वैकल्पिक हैं। इसलिए \(2^2=4\) उपसमुच्चय होंगे।
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