यदि \(U={1,2,\ldots,60}\), \(A={x:x\in U,;2\mid x}\) और \(B={x:x\in U,;3\mid x}\) हैं, तो \(|A\cup B|\) कितना है?
If \(U={1,2,\ldots,60}\), \(A={x:x\in U,;2\mid x}\) and \(B={x:x\in U,;3\mid x}\), what is \(|A\cup B|\)?
#sets
#union
#intersection
#divisibility
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A (40)
B (45)
C (50)
D (30)
Explanation opens after your attempt
Step 1
Concept
Here \(|A\cup B|=30+20-10=40\) because (10) numbers are divisible by (6). In exams, apply inclusion-exclusion first.
Step 2
Why this answer is correct
The correct answer is A. (40). Here \(|A\cup B|=30+20-10=40\) because (10) numbers are divisible by (6). In exams, apply inclusion-exclusion first.
Step 3
Exam Tip
\(|A\cup B|=30+20-10=40\) क्योंकि (6) से विभाज्य (10) संख्याएँ हैं। परीक्षा में समावेशन-अपवर्जन सूत्र पहले लगाएँ।
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यदि (A=[-2,5]) और (B=(1,7]) हैं, तो \(A\setminus B\) क्या है?
If (A=[-2,5]) and (B=(1,7]), what is \(A\setminus B\)?
#sets
#intervals
#difference
#expert
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A ([-2,1])
B ([-2,1))
C ((1,5])
D ([-2,7])
Explanation opens after your attempt
Correct Answer
A. ([-2,1])
Step 1
Concept
Since (1) is not included in (B), elements after (1) are removed from (A), leaving ([-2,1]). In difference questions, check open and closed endpoints carefully.
Step 2
Why this answer is correct
The correct answer is A. ([-2,1]). Since (1) is not included in (B), elements after (1) are removed from (A), leaving ([-2,1]). In difference questions, check open and closed endpoints carefully.
Step 3
Exam Tip
(B) में (1) शामिल नहीं है, इसलिए (A) से (1) के बाद वाले तत्व हटेंगे और ([-2,1]) बचेगा। अंतर निकालते समय खुले और बंद सिरों पर ध्यान दें।
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यदि \(A\cup B=A\cap B\) है, तो (A) और (B) के बारे में सही निष्कर्ष क्या है?
If \(A\cup B=A\cap B\), which conclusion about (A) and (B) is correct?
#sets
#union
#intersection
#equality
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A (A=B)
B \(A=\varnothing\) और \(B\ne\varnothing\) / \(A=\varnothing\) and \(B\ne\varnothing\)
C \(A\cap B=\varnothing\)
D \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
Explanation opens after your attempt
Step 1
Concept
Any element in \(A\cup B\) must also be in \(A\cap B\), so both sets have the same elements. Element-wise reasoning is safest here.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Any element in \(A\cup B\) must also be in \(A\cap B\), so both sets have the same elements. Element-wise reasoning is safest here.
Step 3
Exam Tip
यदि कोई तत्व \(A\cup B\) में है, तो वह \(A\cap B\) में भी है, इसलिए दोनों सेटों में समान तत्व हैं। ऐसी स्थितियों में तत्व-आधारित तर्क सबसे सुरक्षित रहता है।
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यदि \(U={1,2,\ldots,90}\), \(A={x:x\in U,;6\mid x}\) और \(B={x:x\in U,;15\mid x}\) हैं, तो \(|A\cup B|\) कितना है?
If \(U={1,2,\ldots,90}\), \(A={x:x\in U,;6\mid x}\) and \(B={x:x\in U,;15\mid x}\), what is \(|A\cup B|\)?
#sets
#union
#intersection
#divisibility
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A (18)
B (21)
C (24)
D (27)
Explanation opens after your attempt
Step 1
Concept
Here (|A|=15), (|B|=6), and \(|A\cap B|=3\) because common multiples are multiples of (30). So \(|A\cup B|=15+6-3=18\).
Step 2
Why this answer is correct
The correct answer is A. (18). Here (|A|=15), (|B|=6), and \(|A\cap B|=3\) because common multiples are multiples of (30). So \(|A\cup B|=15+6-3=18\).
Step 3
Exam Tip
(|A|=15), (|B|=6) और \(|A\cap B|=3\) क्योंकि सामान्य गुणज (30) के होंगे। अतः \(|A\cup B|=15+6-3=18\) है।
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यदि (|A|=28), (|B|=35) और \(|A\cup B|=50\) है, तो \(|A\setminus B|\) कितना होगा?
If (|A|=28), (|B|=35) and \(|A\cup B|=50\), what is \(|A\setminus B|\)?
#sets
#difference
#cardinality
#union
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A (15)
B (13)
C (22)
D (7)
Explanation opens after your attempt
Step 1
Concept
Here \(|A\cap B|=28+35-50=13\), so \(|A\setminus B|=28-13=15\). Always find the intersection before the difference.
Step 2
Why this answer is correct
The correct answer is B. (13). Here \(|A\cap B|=28+35-50=13\), so \(|A\setminus B|=28-13=15\). Always find the intersection before the difference.
Step 3
Exam Tip
\(|A\cap B|=28+35-50=13\), इसलिए \(|A\setminus B|=28-13=15\) नहीं बल्कि (15) है। पहले प्रतिच्छेद निकालना न भूलें।
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कौन सा कथन सभी सेटों (A), (B), (C) के लिए सदैव सत्य है?
Which statement is always true for all sets (A), (B), (C)?
#sets
#set-identities
#difference
#union
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A (\(A\cup B\)\setminus C=\(A\setminus C\)\cup\(B\setminus C\))
B (A\setminus\(B\cup C\)=\(A\setminus B\)\cup\(A\setminus C\))
C (A\cap\(B\cup C\)=\(A\cup B\)\cap\(A\cup C\))
D (A\cup\(B\cap C\)=\(A\cap B\)\cup\(A\cap C\))
Explanation opens after your attempt
Correct Answer
A. (\(A\cup B\)\setminus C=\(A\setminus C\)\cup\(B\setminus C\))
Step 1
Concept
Removing (C) from \(A\cup B\) leaves elements that are in (A) outside (C) or in (B) outside (C). Recognizing distribution laws saves time in exams.
Step 2
Why this answer is correct
The correct answer is A. (\(A\cup B\)\setminus C=\(A\setminus C\)\cup\(B\setminus C\)). Removing (C) from \(A\cup B\) leaves elements that are in (A) outside (C) or in (B) outside (C). Recognizing distribution laws saves time in exams.
Step 3
Exam Tip
(C) को \(A\cup B\) से हटाने पर वही तत्व बचते हैं जो (A) में (C) से बाहर या (B) में (C) से बाहर हैं। वितरण नियम को पहचानना परीक्षा में समय बचाता है।
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यदि \(A\setminus B=A\) है, तो निम्न में से कौन सा कथन अवश्य सत्य है?
If \(A\setminus B=A\), which statement must be true?
#sets
#difference
#disjoint
#logic
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A \(A\cap B=\varnothing\)
B \(A\subseteq B\)
C \(B\subseteq A\)
D \(A\cup B=A\)
Explanation opens after your attempt
Correct Answer
A. \(A\cap B=\varnothing\)
Step 1
Concept
The condition means no element of (A) is removed by (B). Hence \(A\cap B=\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(A\cap B=\varnothing\). The condition means no element of (A) is removed by (B). Hence \(A\cap B=\varnothing\).
Step 3
Exam Tip
\(A\setminus B=A\) का अर्थ है कि (A) का कोई भी तत्व (B) में नहीं गया। अतः \(A\cap B=\varnothing\) है।
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एक कक्षा में (120) विद्यार्थियों में से (72) विद्यार्थी गणित सेट (M) में, (64) विद्यार्थी भौतिकी सेट (P) में और (28) विद्यार्थी दोनों में हैं। न तो (M) में और न ही (P) में कितने विद्यार्थी हैं?
In a class of (120) students, (72) students are in mathematics set (M), (64) students are in physics set (P), and (28) students are in both. How many students are in neither (M) nor (P)?
#sets
#word-problem
#union
#complement
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A (12)
B (16)
C (20)
D (28)
Explanation opens after your attempt
Step 1
Concept
Here \(|M\cup P|=72+64-28=108\), so students outside both sets are (120-108=12). First find the union, then subtract from the universal set.
Step 2
Why this answer is correct
The correct answer is A. (12). Here \(|M\cup P|=72+64-28=108\), so students outside both sets are (120-108=12). First find the union, then subtract from the universal set.
Step 3
Exam Tip
\(|M\cup P|=72+64-28=108\), इसलिए बाहर के विद्यार्थी (120-108=12) हैं। पहले संघ निकालें, फिर सार्वत्रिक सेट से घटाएँ।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{2,4,6,8\}\) और \(C=\{1,4,7,8\}\) हैं, तो (\(A\cup B\)\cap C) क्या है?
If \(A=\{1,2,3,4,5\}\), \(B=\{2,4,6,8\}\) and \(C=\{1,4,7,8\}\), what is (\(A\cup B\)\cap C)?
#sets
#union
#intersection
#brackets
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A ({1,4,8})
B ({4,8})
C ({1,4,7,8})
D ({2,4,8})
Explanation opens after your attempt
Correct Answer
A. ({1,4,8})
Step 1
Concept
First \(A\cup B={1,2,3,4,5,6,8}\), then common elements with (C) are ({1,4,8}). Always follow the brackets.
Step 2
Why this answer is correct
The correct answer is A. ({1,4,8}). First \(A\cup B={1,2,3,4,5,6,8}\), then common elements with (C) are ({1,4,8}). Always follow the brackets.
Step 3
Exam Tip
पहले \(A\cup B={1,2,3,4,5,6,8}\) है, फिर (C) के साथ सामान्य तत्व ({1,4,8}) हैं। कोष्ठक का क्रम हमेशा ध्यान रखें।
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यदि \(A\subseteq B\), तो (A\cup\(B\setminus A\)) किसके बराबर है?
If \(A\subseteq B\), then (A\cup\(B\setminus A\)) is equal to which set?
#sets
#subset
#difference
#union
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A (B)
B (A)
C \(A\cap B\)
D \(B\setminus A\)
Explanation opens after your attempt
Step 1
Concept
When \(A\subseteq B\), the elements of (B) split into (A) and \(B\setminus A\). Therefore their union is (B).
Step 2
Why this answer is correct
The correct answer is A. (B). When \(A\subseteq B\), the elements of (B) split into (A) and \(B\setminus A\). Therefore their union is (B).
Step 3
Exam Tip
जब \(A\subseteq B\) हो, तो (B) के तत्व (A) और \(B\setminus A\) में बँट जाते हैं। इसलिए उनका संघ (B) है।
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यदि (A\cap\(B\setminus C\)=\varnothing) है, तो कौन सा समावेशन अवश्य सत्य है?
If (A\cap\(B\setminus C\)=\varnothing), which inclusion must be true?
#sets
#difference
#intersection
#inclusion
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A \(A\cap B\subseteq C\)
B \(A\cup B\subseteq C\)
C \(B\subseteq A\cup C\)
D \(C\subseteq A\cap B\)
Explanation opens after your attempt
Correct Answer
A. \(A\cap B\subseteq C\)
Step 1
Concept
Any common element of (A) and (B) cannot lie outside (C). Hence \(A\cap B\subseteq C\).
Step 2
Why this answer is correct
The correct answer is A. \(A\cap B\subseteq C\). Any common element of (A) and (B) cannot lie outside (C). Hence \(A\cap B\subseteq C\).
Step 3
Exam Tip
(A) और (B) में जो भी सामान्य तत्व है, वह (C) के बाहर नहीं हो सकता। इसलिए \(A\cap B\subseteq C\) है।
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यदि \(A={x:x\in\mathbb{Z},;-3\le x\le 4}\) और \(B={x:x\in\mathbb{Z},;x^2<10}\), तो \(A\setminus B\) क्या है?
If \(A={x:x\in\mathbb{Z},;-3\le x\le 4}\) and \(B={x:x\in\mathbb{Z},;x^2<10}\), what is \(A\setminus B\)?
#sets
#set-builder
#integers
#difference
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A ({4})
B ({-3,4})
C ({-4,4})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Here \(B=\{-3,-2,-1,0,1,2,3\}\), and only (4) remains extra in (A). Build each set first, then take the difference.
Step 2
Why this answer is correct
The correct answer is A. ({4}). Here \(B=\{-3,-2,-1,0,1,2,3\}\), and only (4) remains extra in (A). Build each set first, then take the difference.
Step 3
Exam Tip
\(B=\{-3,-2,-1,0,1,2,3\}\) है और (A) में अतिरिक्त केवल (4) है। असमानता से सेट बनाकर फिर अंतर लें।
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यदि (|A|=12), (|B|=18), (|C|=20), \(|A\cap B|=5\), \(|B\cap C|=7\), \(|C\cap A|=4\), \(|A\cap B\cap C|=2\), तो \(|A\cup B\cup C|\) क्या है?
If (|A|=12), (|B|=18), (|C|=20), \(|A\cap B|=5\), \(|B\cap C|=7\), \(|C\cap A|=4\), \(|A\cap B\cap C|=2\), what is \(|A\cup B\cup C|\)?
#sets
#three-sets
#inclusion-exclusion
#union
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A (36)
B (38)
C (40)
D (42)
Explanation opens after your attempt
Step 1
Concept
Here \(|A\cup B\cup C|=12+18+20-5-7-4+2=36\). For three sets, remember to add the triple intersection at the end.
Step 2
Why this answer is correct
The correct answer is A. (36). Here \(|A\cup B\cup C|=12+18+20-5-7-4+2=36\). For three sets, remember to add the triple intersection at the end.
Step 3
Exam Tip
\(|A\cup B\cup C|=12+18+20-5-7-4+2=36\) है। तीन सेटों में अंतिम \(+|A\cap B\cap C|\) जोड़ना याद रखें।
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यदि \(A\cap B=A\) और \(A\cup B=B\), तो निम्न में से कौन सा संबंध सही है?
If \(A\cap B=A\) and \(A\cup B=B\), which relation is correct?
#sets
#subset
#union
#intersection
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A \(A\subseteq B\)
B \(B\subseteq A\)
C \(A\cap B=\varnothing\)
D \(A=B^c\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq B\)
Step 1
Concept
Both conditions show that every element of (A) lies in (B). Hence \(A\subseteq B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\). Both conditions show that every element of (A) lies in (B). Hence \(A\subseteq B\).
Step 3
Exam Tip
दोनों शर्तें यही बताती हैं कि (A) के सभी तत्व (B) में हैं। इसलिए \(A\subseteq B\) है।
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यदि \(A=\{2,3,5,7,11\}\) और \(B={x:x\in A,;x+1\in A}\), तो \(A\setminus B\) क्या है?
If \(A=\{2,3,5,7,11\}\) and \(B={x:x\in A,;x+1\in A}\), what is \(A\setminus B\)?
#sets
#set-builder
#difference
#prime-numbers
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A ({3,5,7,11})
B ({2,5,7,11})
C ({2,3})
D ({11})
Explanation opens after your attempt
Correct Answer
A. ({3,5,7,11})
Step 1
Concept
Only (2+1=3) belongs to (A), so \(B=\{2\}\). Therefore \(A\setminus B={3,5,7,11}\).
Step 2
Why this answer is correct
The correct answer is A. ({3,5,7,11}). Only (2+1=3) belongs to (A), so \(B=\{2\}\). Therefore \(A\setminus B={3,5,7,11}\).
Step 3
Exam Tip
केवल (2+1=3) सेट (A) में है, इसलिए \(B=\{2\}\) है। अतः \(A\setminus B={3,5,7,11}\) है।
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यदि (A\setminus\(B\cup C\)) को केवल (A), (B), (C), \(\cap\) और अंतर के अर्थ से समझना हो, तो इसमें कौन से तत्व होंगे?
If (A\setminus\(B\cup C\)) is described using only the meaning of (A), (B), (C), \(\cap\) and difference, which elements does it contain?
#sets
#difference
#union
#meaning
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A जो (A) में हैं लेकिन (B) और (C) दोनों में नहीं हैं / Elements in (A) but in neither (B) nor (C)
B जो (A), (B) और (C) तीनों में हैं / Elements in all (A), (B) and (C)
C जो (B) या (C) में हैं लेकिन (A) में नहीं हैं / Elements in (B) or (C) but not in (A)
D जो (A) में हैं और \(B\cup C\) में भी हैं / Elements in (A) and also in \(B\cup C\)
Explanation opens after your attempt
Correct Answer
A. जो (A) में हैं लेकिन (B) और (C) दोनों में नहीं हैं / Elements in (A) but in neither (B) nor (C)
Step 1
Concept
Removing \(B\cup C\) means not in (B) and not in (C). So it is the part of (A) outside both sets.
Step 2
Why this answer is correct
The correct answer is A. जो (A) में हैं लेकिन (B) और (C) दोनों में नहीं हैं / Elements in (A) but in neither (B) nor (C). Removing \(B\cup C\) means not in (B) and not in (C). So it is the part of (A) outside both sets.
Step 3
Exam Tip
\(B\cup C\) से हटाने का अर्थ है (B) में नहीं और (C) में नहीं। इसलिए यह (A) का वह भाग है जो दोनों से बाहर है।
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यदि \(A=\{1,2,4,8,16\}\), \(B=\{2,4,6,8,10\}\) और \(C=\{4,8,12,16\}\), तो (\(A\cap C\)\setminus B) क्या है?
If \(A=\{1,2,4,8,16\}\), \(B=\{2,4,6,8,10\}\) and \(C=\{4,8,12,16\}\), what is (\(A\cap C\)\setminus B)?
#sets
#intersection
#difference
#numerical
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A ({16})
B ({4,8})
C ({1,16})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Here \(A\cap C={4,8,16}\), and (4,8) are removed by (B). Therefore the result is ({16}).
Step 2
Why this answer is correct
The correct answer is A. ({16}). Here \(A\cap C={4,8,16}\), and (4,8) are removed by (B). Therefore the result is ({16}).
Step 3
Exam Tip
\(A\cap C={4,8,16}\) है और (B) में (4,8) हट जाते हैं। इसलिए शेष ({16}) है।
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यदि (A\triangle B=\(A\setminus B\)\cup\(B\setminus A\)) और (|A|=21), (|B|=19), \(|A\cap B|=8\), तो \(|A\triangle B|\) क्या है?
If (A\triangle B=\(A\setminus B\)\cup\(B\setminus A\)) and (|A|=21), (|B|=19), \(|A\cap B|=8\), what is \(|A\triangle B|\)?
#sets
#symmetric-difference
#cardinality
#expert
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A (24)
B (32)
C (40)
D (16)
Explanation opens after your attempt
Step 1
Concept
Here \(|A\triangle B|=|A|+|B|-2|A\cap B|=21+19-16=24\). In symmetric difference, the common part is removed twice.
Step 2
Why this answer is correct
The correct answer is A. (24). Here \(|A\triangle B|=|A|+|B|-2|A\cap B|=21+19-16=24\). In symmetric difference, the common part is removed twice.
Step 3
Exam Tip
\(|A\triangle B|=|A|+|B|-2|A\cap B|=21+19-16=24\) है। सममित अंतर में सामान्य भाग दो बार घटता है।
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यदि (\(A\cup B\)\setminus\(A\cap B\)=\varnothing), तो (A) और (B) के बारे में क्या सही है?
If (\(A\cup B\)\setminus\(A\cap B\)=\varnothing), what is true about (A) and (B)?
#sets
#symmetric-difference
#equality
#reasoning
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A (A=B)
B \(A\cap B=\varnothing\)
C \(A\cup B=\varnothing\) लेकिन \(A\ne B\) / \(A\cup B=\varnothing\) but \(A\ne B\)
D \(A\subset B\) उचित रूप से / \(A\subset B\) properly
Explanation opens after your attempt
Step 1
Concept
This set contains elements that lie in exactly one of the two sets. If it is empty, then (A) and (B) are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). This set contains elements that lie in exactly one of the two sets. If it is empty, then (A) and (B) are equal.
Step 3
Exam Tip
यह सेट दोनों में से केवल एक में आने वाले तत्वों का सेट है। इसके खाली होने का अर्थ है (A) और (B) समान हैं।
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यदि \(A\setminus B=B\setminus A\), तो कौन सा निष्कर्ष सही है?
If \(A\setminus B=B\setminus A\), which conclusion is correct?
#sets
#difference
#equality
#logic
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A (A=B)
B \(A\cap B=\varnothing\)
C \(A\cup B=\varnothing\)
D \(A\ne B\) अवश्य / \(A\ne B\) necessarily
Explanation opens after your attempt
Step 1
Concept
The sets \(A\setminus B\) and \(B\setminus A\) are always disjoint, so if they are equal, both are empty. Hence (A=B).
Step 2
Why this answer is correct
The correct answer is A. (A=B). The sets \(A\setminus B\) and \(B\setminus A\) are always disjoint, so if they are equal, both are empty. Hence (A=B).
Step 3
Exam Tip
\(A\setminus B\) और \(B\setminus A\) सदैव असंयुक्त होते हैं, इसलिए उनके बराबर होने पर दोनों खाली होंगे। अतः (A=B) है।
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किस पहचान से (A\cap\(B\cup C\)) को सही रूप में लिखा जा सकता है?
Which identity correctly rewrites (A\cap\(B\cup C\))?
#sets
#distributive-law
#union
#intersection
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A (\(A\cap B\)\cup\(A\cap C\))
B (\(A\cup B\)\cap\(A\cup C\))
C (\(A\setminus B\)\cup\(A\setminus C\))
D (\(B\cap C\)\setminus A)
Explanation opens after your attempt
Correct Answer
A. (\(A\cap B\)\cup\(A\cap C\))
Step 1
Concept
Intersection distributes over union, so (A\cap\(B\cup C\)=\(A\cap B\)\cup\(A\cap C\)). Remembering this identity speeds up Venn diagram questions.
Step 2
Why this answer is correct
The correct answer is A. (\(A\cap B\)\cup\(A\cap C\)). Intersection distributes over union, so (A\cap\(B\cup C\)=\(A\cap B\)\cup\(A\cap C\)). Remembering this identity speeds up Venn diagram questions.
Step 3
Exam Tip
प्रतिच्छेद, संघ पर वितरित होता है, इसलिए (A\cap\(B\cup C\)=\(A\cap B\)\cup\(A\cap C\)) है। सूत्र याद करने से वेन आरेख वाले प्रश्न तेज बनते हैं।
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यदि \(A={x:x\in\mathbb{N},;x\le 20,;x अभाज्य है}\) और (B={x:x\in\mathbb{N},;x\le 20,;x विषम है\(}), तो (A\setminus B) क्या है\)?
If \(A={x:x\in\mathbb{N},;x\le 20,;x is prime}\) and (B={x:x\in\mathbb{N},;x\le 20,;x is odd\(}), what is (A\setminus B)\)?
#sets
#set-builder
#prime
#odd
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A ({2})
B \(\varnothing\)
C ({3,5,7,11,13,17,19})
D ({1,2})
Explanation opens after your attempt
Step 1
Concept
Only (2) is an even prime number up to (20). Therefore after removing odd numbers, ({2}) remains.
Step 2
Why this answer is correct
The correct answer is A. ({2}). Only (2) is an even prime number up to (20). Therefore after removing odd numbers, ({2}) remains.
Step 3
Exam Tip
(20) तक केवल (2) ही सम अभाज्य संख्या है। इसलिए विषम संख्याएँ हटाने पर ({2}) बचता है।
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यदि \(A\subseteq C\) और \(B\subseteq C\), तो (\(A\cup B\)\setminus C) क्या होगा?
If \(A\subseteq C\) and \(B\subseteq C\), what is (\(A\cup B\)\setminus C)?
#sets
#subset
#union
#difference
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A \(\varnothing\)
B \(A\cup B\)
C (C\setminus\(A\cup B\))
D \(A\cap B\)
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
Every element of \(A\cup B\) lies in (C), so removing (C) leaves nothing. Hence the result is \(\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). Every element of \(A\cup B\) lies in (C), so removing (C) leaves nothing. Hence the result is \(\varnothing\).
Step 3
Exam Tip
\(A\cup B\) का हर तत्व (C) में है, इसलिए (C) हटाने पर कुछ नहीं बचेगा। अतः परिणाम \(\varnothing\) है।
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यदि \(U={1,2,\ldots,100}\), \(A={x:x\in U,;4\mid x}\) और \(B={x:x\in U,;10\mid x}\), तो \(|A\cap B|\) क्या है?
If \(U={1,2,\ldots,100}\), \(A={x:x\in U,;4\mid x}\) and \(B={x:x\in U,;10\mid x}\), what is \(|A\cap B|\)?
#sets
#intersection
#divisibility
#lcm
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A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
Common numbers must be divisible by the least common multiple (20) of (4) and (10). There are (5) such numbers up to (100).
Step 2
Why this answer is correct
The correct answer is A. (5). Common numbers must be divisible by the least common multiple (20) of (4) and (10). There are (5) such numbers up to (100).
Step 3
Exam Tip
सामान्य संख्याएँ (4) और (10) के लघुत्तम समापवर्त्य (20) से विभाज्य होंगी। (100) तक ऐसी (5) संख्याएँ हैं।
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यदि \(A\cup B=A\cup C\) और \(A\cap B=A\cap C\), तो कौन सा निष्कर्ष अवश्य सत्य है?
If \(A\cup B=A\cup C\) and \(A\cap B=A\cap C\), which conclusion must be true?
#sets
#union
#intersection
#equality-proof
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A (B=C)
B (A=B=C)
C \(B\cap C=\varnothing\)
D \(A\setminus B=A\setminus C\) ही पर्याप्त निष्कर्ष है / Only \(A\setminus B=A\setminus C\) is enough
Explanation opens after your attempt
Step 1
Concept
For any element (x), inside (A) and outside (A), membership in (B) and (C) matches. Hence (B=C).
Step 2
Why this answer is correct
The correct answer is A. (B=C). For any element (x), inside (A) and outside (A), membership in (B) and (C) matches. Hence (B=C).
Step 3
Exam Tip
किसी तत्व (x) के लिए (A) के अंदर और बाहर दोनों स्थितियों में (B) और (C) की सदस्यता समान मिलती है। इसलिए (B=C) है।
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यदि \(A=\{a,b,c,d\}\), \(B=\{b,d,e\}\), तो (\(A\setminus B\)\cup\(B\setminus A\)) क्या है?
If \(A=\{a,b,c,d\}\), \(B=\{b,d,e\}\), what is (\(A\setminus B\)\cup\(B\setminus A\))?
#sets
#symmetric-difference
#letters
#difference
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A ({a,c,e})
B ({b,d})
C ({a,b,c,d,e})
D ({a,c})
Explanation opens after your attempt
Correct Answer
A. ({a,c,e})
Step 1
Concept
Here \(A\setminus B={a,c}\) and \(B\setminus A={e}\). Their union is ({a,c,e}).
Step 2
Why this answer is correct
The correct answer is A. ({a,c,e}). Here \(A\setminus B={a,c}\) and \(B\setminus A={e}\). Their union is ({a,c,e}).
Step 3
Exam Tip
\(A\setminus B={a,c}\) और \(B\setminus A={e}\) है। उनका संघ ({a,c,e}) है।
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यदि \(A\cap B=\varnothing\), तो \(|A\cup B|\) के लिए सही सूत्र कौन सा है?
If \(A\cap B=\varnothing\), which formula for \(|A\cup B|\) is correct?
#sets
#disjoint
#union
#cardinality
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A (|A|+|B|)
B (|A|+|B|-1)
C (|A|-|B|)
D \(|A\cap B|\)
Explanation opens after your attempt
Correct Answer
A. (|A|+|B|)
Step 1
Concept
Disjoint sets have no repeated common element. Therefore the number of elements in the union is simply (|A|+|B|).
Step 2
Why this answer is correct
The correct answer is A. (|A|+|B|). Disjoint sets have no repeated common element. Therefore the number of elements in the union is simply (|A|+|B|).
Step 3
Exam Tip
असंयुक्त सेटों में कोई दोहराया हुआ तत्व नहीं होता। इसलिए संघ की संख्या सीधे (|A|+|B|) होती है।
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यदि \(A\setminus B={1,4}\), \(A\cap B={2,3}\) और \(B\setminus A={5,6,7}\), तो \(A\cup B\) क्या है?
If \(A\setminus B={1,4}\), \(A\cap B={2,3}\) and \(B\setminus A={5,6,7}\), what is \(A\cup B\)?
#sets
#venn-parts
#union
#difference
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A ({1,2,3,4,5,6,7})
B ({2,3,5,6,7})
C ({1,4,5,6,7})
D ({2,3})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3,4,5,6,7})
Step 1
Concept
The union is made from the three disjoint parts \(A\setminus B\), \(A\cap B\), and \(B\setminus A\). Combining all gives ({1,2,3,4,5,6,7}).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,4,5,6,7}). The union is made from the three disjoint parts \(A\setminus B\), \(A\cap B\), and \(B\setminus A\). Combining all gives ({1,2,3,4,5,6,7}).
Step 3
Exam Tip
संघ तीन असंयुक्त भागों \(A\setminus B\), \(A\cap B\), और \(B\setminus A\) से बनता है। सभी तत्व मिलाकर ({1,2,3,4,5,6,7}) मिलते हैं।
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यदि \(A={x:x\in\mathbb{R},;x^2-5x+6=0}\) और \(B={x:x\in\mathbb{R},;x^2-7x+12=0}\), तो \(A\cap B\) क्या है?
If \(A={x:x\in\mathbb{R},;x^2-5x+6=0}\) and \(B={x:x\in\mathbb{R},;x^2-7x+12=0}\), what is \(A\cap B\)?
#sets
#quadratic-roots
#intersection
#set-builder
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A ({3})
B ({2,3,4})
C ({2,4})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
First \(A=\{2,3\}\) and \(B=\{3,4\}\). The only common element is (3).
Step 2
Why this answer is correct
The correct answer is A. ({3}). First \(A=\{2,3\}\) and \(B=\{3,4\}\). The only common element is (3).
Step 3
Exam Tip
पहले \(A=\{2,3\}\) और \(B=\{3,4\}\) मिलते हैं। सामान्य तत्व केवल (3) है।
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यदि (A) और (B) सीमित सेट हैं तथा \(|A\setminus B|=9\), \(|B\setminus A|=4\), \(|A\cap B|=11\), तो \(|A\cup B|\) क्या है?
If (A) and (B) are finite sets and \(|A\setminus B|=9\), \(|B\setminus A|=4\), \(|A\cap B|=11\), what is \(|A\cup B|\)?
#sets
#venn-counting
#union
#cardinality
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A (24)
B (20)
C (15)
D (13)
Explanation opens after your attempt
Step 1
Concept
The union is the sum of three separate parts: (9+4+11=24). In a Venn diagram, add the disjoint regions.
Step 2
Why this answer is correct
The correct answer is A. (24). The union is the sum of three separate parts: (9+4+11=24). In a Venn diagram, add the disjoint regions.
Step 3
Exam Tip
संघ तीन अलग भागों का योग है: (9+4+11=24)। वेन आरेख में अलग क्षेत्रों की गिनती जोड़ें।
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यदि \(A\cup B=A\) और \(A\cap B=B\), तो कौन सा कथन सही है?
If \(A\cup B=A\) and \(A\cap B=B\), which statement is correct?
#sets
#subset
#union
#intersection
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A \(B\subseteq A\)
B \(A\subseteq B\)
C \(A\cap B=\varnothing\)
D \(A=B^c\)
Explanation opens after your attempt
Correct Answer
A. \(B\subseteq A\)
Step 1
Concept
Both conditions mean every element of (B) lies in (A). Therefore \(B\subseteq A\).
Step 2
Why this answer is correct
The correct answer is A. \(B\subseteq A\). Both conditions mean every element of (B) lies in (A). Therefore \(B\subseteq A\).
Step 3
Exam Tip
इन दोनों शर्तों का अर्थ है कि (B) का हर तत्व (A) में है। इसलिए \(B\subseteq A\) है।
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यदि \(A=\{1,3,5,7,9\}\), \(B=\{0,3,6,9\}\), \(C=\{3,4,5,9\}\), तो (A\cap\(B\cup C\)) क्या है?
If \(A=\{1,3,5,7,9\}\), \(B=\{0,3,6,9\}\), \(C=\{3,4,5,9\}\), what is (A\cap\(B\cup C\))?
#sets
#union
#intersection
#calculation
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A ({3,5,9})
B ({3,9})
C ({1,5,7})
D ({0,3,4,5,6,9})
Explanation opens after your attempt
Correct Answer
A. ({3,5,9})
Step 1
Concept
Here \(B\cup C={0,3,4,5,6,9}\), and the common elements with (A) are ({3,5,9}). Find the inner union first.
Step 2
Why this answer is correct
The correct answer is A. ({3,5,9}). Here \(B\cup C={0,3,4,5,6,9}\), and the common elements with (A) are ({3,5,9}). Find the inner union first.
Step 3
Exam Tip
\(B\cup C={0,3,4,5,6,9}\) है और (A) के साथ सामान्य तत्व ({3,5,9}) हैं। पहले अंदर का संघ निकालें।
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यदि \(A\setminus C=B\setminus C\) और \(A\cap C=B\cap C\), तो (A) और (B) के बारे में क्या निष्कर्ष है?
If \(A\setminus C=B\setminus C\) and \(A\cap C=B\cap C\), what is the conclusion about (A) and (B)?
#sets
#partition
#difference
#intersection
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A (A=B)
B \(A\cap B=\varnothing\)
C \(A\cup B=C\)
D \(C\subseteq A\setminus B\)
Explanation opens after your attempt
Step 1
Concept
Any set (A) can be split into (\(A\setminus C\)) and (\(A\cap C\)). Since both parts match those of (B), (A=B).
Step 2
Why this answer is correct
The correct answer is A. (A=B). Any set (A) can be split into (\(A\setminus C\)) and (\(A\cap C\)). Since both parts match those of (B), (A=B).
Step 3
Exam Tip
किसी भी सेट (A) को (\(A\setminus C\)) और (\(A\cap C\)) में बाँटा जा सकता है। दोनों भाग (B) के समान हैं, इसलिए (A=B) है।
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यदि \(A\cap B\subseteq C\), तो कौन सा सेट अवश्य खाली है?
If \(A\cap B\subseteq C\), which set must be empty?
#sets
#subset
#difference
#empty-set
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A (\(A\cap B\)\setminus C)
B (C\setminus\(A\cap B\))
C \(A\setminus C\)
D \(B\setminus C\)
Explanation opens after your attempt
Correct Answer
A. (\(A\cap B\)\setminus C)
Step 1
Concept
If every element of \(A\cap B\) lies in (C), then none of its elements lies outside (C). Hence (\(A\cap B\)\setminus C=\varnothing).
Step 2
Why this answer is correct
The correct answer is A. (\(A\cap B\)\setminus C). If every element of \(A\cap B\) lies in (C), then none of its elements lies outside (C). Hence (\(A\cap B\)\setminus C=\varnothing).
Step 3
Exam Tip
यदि \(A\cap B\) का हर तत्व (C) में है, तो (C) के बाहर इसका कोई तत्व नहीं होगा। इसलिए (\(A\cap B\)\setminus C=\varnothing) है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{3,4,5,6\}\), तो (\mathcal{P}\(A\cap B\)\cap\mathcal{P}\(A\setminus B\)) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{3,4,5,6\}\), what is (\mathcal{P}\(A\cap B\)\cap\mathcal{P}\(A\setminus B\))?
#sets
#power-set
#intersection
#difference
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A \({\varnothing}\)
B \(\varnothing\)
C ({{1},{2}})
D ({{3},{4}})
Explanation opens after your attempt
Correct Answer
A. \({\varnothing}\)
Step 1
Concept
Here \(A\cap B={3,4}\) and \(A\setminus B={1,2}\) are disjoint. The only common subset in their power sets is \(\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \({\varnothing}\). Here \(A\cap B={3,4}\) and \(A\setminus B={1,2}\) are disjoint. The only common subset in their power sets is \(\varnothing\).
Step 3
Exam Tip
\(A\cap B={3,4}\) और \(A\setminus B={1,2}\) असंयुक्त हैं। इनके पावर सेटों में सामान्य उपसमुच्चय केवल \(\varnothing\) है।
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यदि \(A\cup B=A\cup C\) है, तो कौन सा अतिरिक्त कथन (B=C) सिद्ध करने के लिए पर्याप्त है?
If \(A\cup B=A\cup C\), which additional statement is sufficient to prove (B=C)?
#sets
#equality-proof
#union
#intersection
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A \(A\cap B=A\cap C\)
B \(B\cap C=\varnothing\)
C \(A\setminus B=A\)
D \(A\cup B=U\)
Explanation opens after your attempt
Correct Answer
A. \(A\cap B=A\cap C\)
Step 1
Concept
Equal unions and equal intersections make membership in (B) and (C) identical in every case. Therefore (B=C).
Step 2
Why this answer is correct
The correct answer is A. \(A\cap B=A\cap C\). Equal unions and equal intersections make membership in (B) and (C) identical in every case. Therefore (B=C).
Step 3
Exam Tip
समान संघ और समान प्रतिच्छेद मिलकर (B) और (C) की सदस्यता को हर स्थिति में समान कर देते हैं। इसलिए (B=C) सिद्ध होता है।
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यदि \(A\setminus B=A\cap C\) और \(A\cap B=A\setminus C\), तो \(A\cap B\cap C\) क्या होगा?
If \(A\setminus B=A\cap C\) and \(A\cap B=A\setminus C\), what is \(A\cap B\cap C\)?
#sets
#intersection
#difference
#logic
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A \(\varnothing\)
B (A)
C \(B\cap C\)
D (A\setminus\(B\cup C\))
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
Since \(A\cap B=A\setminus C\), no element of \(A\cap B\) lies in (C). Hence \(A\cap B\cap C=\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). Since \(A\cap B=A\setminus C\), no element of \(A\cap B\) lies in (C). Hence \(A\cap B\cap C=\varnothing\).
Step 3
Exam Tip
\(A\cap B=A\setminus C\) होने से \(A\cap B\) का कोई तत्व (C) में नहीं है। इसलिए \(A\cap B\cap C=\varnothing\) है।
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यदि \(A={x:x\in\mathbb{Z},;|x|\le 4}\) और \(B={x:x\in\mathbb{Z},;x^2\le 9}\), तो \(A\setminus B\) क्या है?
If \(A={x:x\in\mathbb{Z},;|x|\le 4}\) and \(B={x:x\in\mathbb{Z},;x^2\le 9}\), what is \(A\setminus B\)?
#sets
#integers
#absolute-value
#difference
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A ({-4,4})
B ({-3,3})
C ({-4,-3,3,4})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({-4,4})
Step 1
Concept
Here \(A=\{-4,-3,-2,-1,0,1,2,3,4\}\) and \(B=\{-3,-2,-1,0,1,2,3\}\). The difference leaves ({-4,4}).
Step 2
Why this answer is correct
The correct answer is A. ({-4,4}). Here \(A=\{-4,-3,-2,-1,0,1,2,3,4\}\) and \(B=\{-3,-2,-1,0,1,2,3\}\). The difference leaves ({-4,4}).
Step 3
Exam Tip
\(A=\{-4,-3,-2,-1,0,1,2,3,4\}\) और \(B=\{-3,-2,-1,0,1,2,3\}\) है। अंतर में ({-4,4}) बचता है।
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कौन सा कथन (A\setminus\(B\setminus C\)) के बराबर है?
Which expression is equal to (A\setminus\(B\setminus C\))?
#sets
#difference
#identity
#advanced
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A (\(A\setminus B\)\cup\(A\cap C\))
B (A\setminus\(B\cup C\))
C (A\cap\(B\setminus C\))
D (\(A\cup B\)\setminus C)
Explanation opens after your attempt
Correct Answer
A. (\(A\setminus B\)\cup\(A\cap C\))
Step 1
Concept
The set \(B\setminus C\) has elements in (B) but not in (C). Removing these from (A) leaves elements of (A) outside (B) or inside (C).
Step 2
Why this answer is correct
The correct answer is A. (\(A\setminus B\)\cup\(A\cap C\)). The set \(B\setminus C\) has elements in (B) but not in (C). Removing these from (A) leaves elements of (A) outside (B) or inside (C).
Step 3
Exam Tip
\(B\setminus C\) में (B) के वे तत्व हैं जो (C) में नहीं हैं। (A) से इन्हें हटाने पर (A) में या तो (B) से बाहर वाले या (C) में रहने वाले तत्व बचते हैं।
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यदि (n\(A\cup B\)=n(A)+n(B)) और (n(A)=0), तो \(A\cap B\) क्या है?
If (n\(A\cup B\)=n(A)+n(B)) and (n(A)=0), what is \(A\cap B\)?
#sets
#empty-set
#intersection
#cardinality
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A \(\varnothing\)
B (B)
C \(A\cup B\)
D (U)
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
Since (n(A)=0), \(A=\varnothing\), so \(A\cap B=\varnothing\). The intersection of the empty set with any set is empty.
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). Since (n(A)=0), \(A=\varnothing\), so \(A\cap B=\varnothing\). The intersection of the empty set with any set is empty.
Step 3
Exam Tip
(n(A)=0) से \(A=\varnothing\) है, इसलिए \(A\cap B=\varnothing\) होगा। खाली सेट का किसी भी सेट से प्रतिच्छेद खाली होता है।
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यदि \(A\cap B=A\setminus B\), तो (A) के बारे में कौन सा निष्कर्ष सही है?
If \(A\cap B=A\setminus B\), which conclusion about (A) is correct?
#sets
#intersection
#difference
#empty-set
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A \(A=\varnothing\)
B \(B=\varnothing\)
C (A=B)
D \(A\cup B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \(A=\varnothing\)
Step 1
Concept
The sets \(A\cap B\) and \(A\setminus B\) are always disjoint. If they are equal, both are empty, so (A) is empty.
Step 2
Why this answer is correct
The correct answer is A. \(A=\varnothing\). The sets \(A\cap B\) and \(A\setminus B\) are always disjoint. If they are equal, both are empty, so (A) is empty.
Step 3
Exam Tip
\(A\cap B\) और \(A\setminus B\) सदैव असंयुक्त होते हैं। बराबर होने पर दोनों खाली होंगे, इसलिए (A) भी खाली है।
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यदि (A), (B), (C) ऐसे सेट हैं कि \(A\subseteq B\) और \(B\cap C=\varnothing\), तो \(A\cap C\) क्या होगा?
If sets (A), (B), (C) satisfy \(A\subseteq B\) and \(B\cap C=\varnothing\), what is \(A\cap C\)?
#sets
#subset
#disjoint
#intersection
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A \(\varnothing\)
B (A)
C (C)
D \(B\setminus A\)
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
All elements of (A) are in (B), and (B) has no common element with (C). Therefore \(A\cap C=\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). All elements of (A) are in (B), and (B) has no common element with (C). Therefore \(A\cap C=\varnothing\).
Step 3
Exam Tip
(A) के सभी तत्व (B) में हैं और (B) का (C) से कोई सामान्य तत्व नहीं है। इसलिए \(A\cap C=\varnothing\) है।
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यदि \(A={x:x\in\mathbb{N},;x\le 12}\), \(B={x:x\in A,;x सम है}\), और (C={x:\(x\in A\),;x अभाज्य है\(}), तो (B\cup C) में कितने तत्व हैं\)?
If \(A={x:x\in\mathbb{N},;x\le 12}\), \(B={x:x\in A,;x is even}\), and (C={x:\(x\in A\),;x is prime\(}), how many elements are in (B\cup C)\)?
#sets
#union
#counting
#prime-even
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A (9)
B (10)
C (8)
D (7)
Explanation opens after your attempt
Step 1
Concept
Here \(B=\{2,4,6,8,10,12\}\) and \(C=\{2,3,5,7,11\}\). Counting common (2) once gives (9) elements.
Step 2
Why this answer is correct
The correct answer is A. (9). Here \(B=\{2,4,6,8,10,12\}\) and \(C=\{2,3,5,7,11\}\). Counting common (2) once gives (9) elements.
Step 3
Exam Tip
\(B=\{2,4,6,8,10,12\}\) और \(C=\{2,3,5,7,11\}\) हैं। सामान्य (2) को एक बार गिनकर कुल (9) तत्व मिलते हैं।
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यदि \(A\cup B=A\setminus B\), तो (B) के बारे में क्या सही है?
If \(A\cup B=A\setminus B\), what is true about (B)?
#sets
#union
#difference
#logic
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A \(B=\varnothing\)
B \(A=\varnothing\)
C (A=B)
D \(B\subseteq A\) और \(B\ne\varnothing\) / \(B\subseteq A\) and \(B\ne\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \(B=\varnothing\)
Step 1
Concept
The right side contains no element of (B), while the left side contains all elements of (B). Equality is possible only when \(B=\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(B=\varnothing\). The right side contains no element of (B), while the left side contains all elements of (B). Equality is possible only when \(B=\varnothing\).
Step 3
Exam Tip
दायाँ पक्ष (B) का कोई तत्व नहीं रखता, पर बायाँ पक्ष (B) के सभी तत्व रखता है। समानता तभी संभव है जब \(B=\varnothing\) हो।
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यदि \(A\cap B=A\cup B\cup C\), तो (C) के बारे में कौन सा निष्कर्ष अवश्य सही है?
If \(A\cap B=A\cup B\cup C\), which conclusion about (C) must be true?
#sets
#advanced-equality
#union
#intersection
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A \(C\subseteq A\cap B\) और (A=B) / \(C\subseteq A\cap B\) and (A=B)
B \(C=\varnothing\) ही अवश्य / Only \(C=\varnothing\) necessarily
C \(A\cap B=\varnothing\)
D \(C\supseteq A\cup B\)
Explanation opens after your attempt
Correct Answer
A. \(C\subseteq A\cap B\) और (A=B) / \(C\subseteq A\cap B\) and (A=B)
Step 1
Concept
Since \(A\cap B\subseteq A\cup B\), equality forces \(A\cup B=A\cap B\), so (A=B), and also \(C\subseteq A\cap B\).
Step 2
Why this answer is correct
The correct answer is A. \(C\subseteq A\cap B\) और (A=B) / \(C\subseteq A\cap B\) and (A=B). Since \(A\cap B\subseteq A\cup B\), equality forces \(A\cup B=A\cap B\), so (A=B), and also \(C\subseteq A\cap B\).
Step 3
Exam Tip
क्योंकि \(A\cap B\subseteq A\cup B\) और समानता में \(A\cup B\) भी उसी के बराबर होना चाहिए, इसलिए (A=B) और \(C\subseteq A\cap B\) है।
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यदि \(A=\{1,2,3\}\), \(B=\{3,4\}\), और \(C=\{2,3,5\}\), तो (\(A\cup B\)\setminus C) क्या है?
If \(A=\{1,2,3\}\), \(B=\{3,4\}\), and \(C=\{2,3,5\}\), what is (\(A\cup B\)\setminus C)?
#sets
#union
#difference
#basic-to-expert
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A ({1,4})
B ({2,3})
C ({1,2,4})
D ({5})
Explanation opens after your attempt
Correct Answer
A. ({1,4})
Step 1
Concept
Here \(A\cup B={1,2,3,4}\), and (2,3) are removed by (C). So ({1,4}) remains.
Step 2
Why this answer is correct
The correct answer is A. ({1,4}). Here \(A\cup B={1,2,3,4}\), and (2,3) are removed by (C). So ({1,4}) remains.
Step 3
Exam Tip
\(A\cup B={1,2,3,4}\) है और (C) से (2,3) हटते हैं। इसलिए ({1,4}) बचता है।
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यदि \(|A\cup B|=70\), \(|A\setminus B|=25\) और \(|B\setminus A|=30\), तो \(|A\cap B|\) क्या है?
If \(|A\cup B|=70\), \(|A\setminus B|=25\) and \(|B\setminus A|=30\), what is \(|A\cap B|\)?
#sets
#cardinality
#venn-regions
#intersection
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A (15)
B (20)
C (25)
D (10)
Explanation opens after your attempt
Step 1
Concept
The union is the sum of \(A\setminus B\), \(B\setminus A\), and \(A\cap B\). Hence \(|A\cap B|=70-25-30=15\).
Step 2
Why this answer is correct
The correct answer is A. (15). The union is the sum of \(A\setminus B\), \(B\setminus A\), and \(A\cap B\). Hence \(|A\cap B|=70-25-30=15\).
Step 3
Exam Tip
संघ \(A\setminus B\), \(B\setminus A\), और \(A\cap B\) का योग है। अतः \(|A\cap B|=70-25-30=15\) है।
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यदि \(A\setminus B\subseteq C\) और \(A\cap B\subseteq C\), तो कौन सा निष्कर्ष सही है?
If \(A\setminus B\subseteq C\) and \(A\cap B\subseteq C\), which conclusion is correct?
#sets
#decomposition
#subset
#difference
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A \(A\subseteq C\)
B \(B\subseteq C\)
C \(A\cup B\subseteq C\)
D \(C\subseteq A\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq C\)
Step 1
Concept
We have (A=\(A\setminus B\)\cup\(A\cap B\)). Since both parts lie in (C), \(A\subseteq C\).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq C\). We have (A=\(A\setminus B\)\cup\(A\cap B\)). Since both parts lie in (C), \(A\subseteq C\).
Step 3
Exam Tip
(A=\(A\setminus B\)\cup\(A\cap B\)) होता है। दोनों भाग (C) में हैं, इसलिए \(A\subseteq C\) है।
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यदि \(A\cup B=U\) और \(A\setminus B=\varnothing\), तो (B) के बारे में क्या अवश्य सत्य है?
If \(A\cup B=U\) and \(A\setminus B=\varnothing\), what must be true about (B)?
#sets
#subset
#union
#difference
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A (B=U)
B (A=U)
C \(A\cap B=\varnothing\)
D \(B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
From \(A\setminus B=\varnothing\), \(A\subseteq B\), so \(A\cup B=B\). Since \(A\cup B=U\), we get (B=U).
Step 2
Why this answer is correct
The correct answer is A. (B=U). From \(A\setminus B=\varnothing\), \(A\subseteq B\), so \(A\cup B=B\). Since \(A\cup B=U\), we get (B=U).
Step 3
Exam Tip
\(A\setminus B=\varnothing\) से \(A\subseteq B\) है, इसलिए \(A\cup B=B\) होगा। चूँकि \(A\cup B=U\), इसलिए (B=U) है।
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यदि (A), (B), (C) के लिए (\(A\setminus B\)\cap C=A\cap\(C\setminus B\)) की सत्यता जाँची जाए, तो सही कथन कौन सा है?
If the truth of (\(A\setminus B\)\cap C=A\cap\(C\setminus B\)) is checked for sets (A), (B), (C), which statement is correct?
#sets
#identity
#difference
#intersection
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A यह सदैव सत्य है / It is always true
B यह केवल तब सत्य है जब (A=B) / It is true only when (A=B)
C यह केवल तब सत्य है जब \(C=\varnothing\) / It is true only when \(C=\varnothing\)
D यह कभी सत्य नहीं है / It is never true
Explanation opens after your attempt
Correct Answer
A. यह सदैव सत्य है / It is always true
Step 1
Concept
Both sides describe elements that are in (A) and (C) but not in (B). Hence the identity is always true.
Step 2
Why this answer is correct
The correct answer is A. यह सदैव सत्य है / It is always true. Both sides describe elements that are in (A) and (C) but not in (B). Hence the identity is always true.
Step 3
Exam Tip
दोनों पक्ष उन तत्वों को बताते हैं जो (A) और (C) में हैं लेकिन (B) में नहीं हैं। इसलिए पहचान सदैव सत्य है।
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