यदि \(U={1,2,\ldots,30}\), (A) अभाज्य संख्याओं का समुच्चय है और (B) (3) के गुणजों का समुच्चय है, तो (\(A\cup B\)') में कितने अवयव हैं?
If \(U={1,2,\ldots,30}\), (A) is the set of prime numbers and (B) is the set of multiples of (3), then how many elements are in (\(A\cup B\)')?
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#counting
#union
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A (11)
B (12)
C (13)
D (14)
Explanation opens after your attempt
Step 1
Concept
There are (10) primes in (A), (10) multiples in (B), and \(A\cap B={3}\), so \(|A\cup B|=19\). Hence (|\(A\cup B\)'|=30-19=11); always subtract from (|U|).
Step 2
Why this answer is correct
The correct answer is B. (12). There are (10) primes in (A), (10) multiples in (B), and \(A\cap B={3}\), so \(|A\cup B|=19\). Hence (|\(A\cup B\)'|=30-19=11); always subtract from (|U|).
Step 3
Exam Tip
(A) में (10) अभाज्य हैं, (B) में (10) गुणज हैं और \(A\cap B={3}\), इसलिए \(|A\cup B|=19\)। अतः (|\(A\cup B\)'|=30-19=11) नहीं, ध्यान से (30-19=11) है।
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\(यदि (U={1,2,\ldots,45}), (A={x:x\) 3 से विभाज्य है\(}) और (B={x:x\) 5 से विभाज्य है\(}), तो (|(A\cap B)'|) कितना है\)?
\(If (U={1,2,\ldots,45}), (A={x:x\) is divisible by \(3}) and (B={x:x\) is divisible by \(5}), what is (|(A\cap B)'|)\)?
#sets
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#intersection
#counting
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A (40)
B (41)
C (42)
D (43)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B\) contains multiples of (15), namely (15,30,45), so it has (3) elements. Hence (|\(A\cap B\)'|=45-3=42).
Step 2
Why this answer is correct
The correct answer is C. (42). \(A\cap B\) contains multiples of (15), namely (15,30,45), so it has (3) elements. Hence (|\(A\cap B\)'|=45-3=42).
Step 3
Exam Tip
\(A\cap B\) में (15) के गुणज (15,30,45) हैं, इसलिए उसमें (3) अवयव हैं। अतः (|\(A\cap B\)'|=45-3=42)।
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यदि \(U={x:x\in \mathbb{Z},-5\le x\le 5}\) और \(A={x:x^2\le 9}\), तो (A') क्या होगा?
If \(U={x:x\in \mathbb{Z},-5\le x\le 5}\) and \(A={x:x^2\le 9}\), what is (A')?
#sets
#complement
#integers
#set-builder
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A \({-5,-4,4,5})
B \({-4,4})
C \({-5,5})
D \({-3,-2,-1,0,1,2,3})
Explanation opens after your attempt
Correct Answer
A. \({-5,-4,4,5})
Step 1
Concept
Here \(A=\{-3,-2,-1,0,1,2,3\}\), so the remaining elements of (U) form (A'). In exams, always find complement inside (U).
Step 2
Why this answer is correct
The correct answer is A. \({-5,-4,4,5}). Here \(A=\{-3,-2,-1,0,1,2,3\}\), so the remaining elements of (U) form (A'). In exams, always find complement inside (U).
Step 3
Exam Tip
\(A=\{-3,-2,-1,0,1,2,3\}\) है, इसलिए (U) के बचे अवयव (A') होंगे। परीक्षा में पूरक हमेशा (U) के अंदर ही निकालें।
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यदि \(U=\mathbb{R}\) और \(A={x:-2<x\le 6}\), तो (A') क्या होगा?
If \(U=\mathbb{R}\) and \(A={x:-2<x\le 6}\), what is (A')?
#sets
#complement
#intervals
#real-numbers
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A \((-\infty,-2]\cup\(6,\infty\))
B \(\(-\infty,-2\)\cup[6,\infty))
C \([-2,6])
D \((-2,6])
Explanation opens after your attempt
Correct Answer
A. \((-\infty,-2]\cup\(6,\infty\))
Step 1
Concept
(A) does not include (-2) but includes (6), so the complement includes (-2) but excludes (6). The correct answer is ((-\infty,-2]\cup\(6,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. \((-\infty,-2]\cup\(6,\infty\)). (A) does not include (-2) but includes (6), so the complement includes (-2) but excludes (6). The correct answer is ((-\infty,-2]\cup\(6,\infty\)).
Step 3
Exam Tip
(A) में (-2) शामिल नहीं और (6) शामिल है, इसलिए पूरक में (-2) आएगा पर (6) नहीं आएगा। सही उत्तर ((-\infty,-2]\cup\(6,\infty\)) है।
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यदि \(A\subseteq B\subseteq U\), तो निम्न में से कौन सा संबंध सदैव सत्य है?
If \(A\subseteq B\subseteq U\), which relation is always true?
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#property
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A \(A'\subseteq B'\)
B \(B'\subseteq A'\)
C (A'=B')
D \(A'\cap B'=U\)
Explanation opens after your attempt
Correct Answer
B. \(B'\subseteq A'\)
Step 1
Concept
The complement reverses inclusion, so \(A\subseteq B\Rightarrow B'\subseteq A'\). Remember this reversal property for hard questions.
Step 2
Why this answer is correct
The correct answer is B. \(B'\subseteq A'\). The complement reverses inclusion, so \(A\subseteq B\Rightarrow B'\subseteq A'\). Remember this reversal property for hard questions.
Step 3
Exam Tip
बड़े समुच्चय का पूरक छोटा होता है, इसलिए \(A\subseteq B\Rightarrow B'\subseteq A'\)। इस उल्टे क्रम को याद रखना उपयोगी है।
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यदि (U) सार्वत्रिक समुच्चय है, तो (\(A\cap B'\)') किसके बराबर है?
If (U) is the universal set, then (\(A\cap B'\)') is equal to what?
#sets
#demorgan
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#algebra
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A \(A'\cup B\)
B \(A'\cap B\)
C \(A\cup B'\)
D \(A\cap B\)
Explanation opens after your attempt
Correct Answer
A. \(A'\cup B\)
Step 1
Concept
By De Morgan's law, (\(A\cap B'\)'=A'\cup (B')'=A'\cup B). First identify the inner complement.
Step 2
Why this answer is correct
The correct answer is A. \(A'\cup B\). By De Morgan's law, (\(A\cap B'\)'=A'\cup (B')'=A'\cup B). First identify the inner complement.
Step 3
Exam Tip
डी मॉर्गन नियम से (\(A\cap B'\)'=A'\cup (B')'=A'\cup B)। पहले अंदर के पूरक को पहचानें।
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\(यदि (U={1,2,\ldots,20}), (A={x:x\) सम है\(}) और (B={x:x\) 5 का गुणज है\(}), तो (A'\cap B') क्या दर्शाता है\)?
\(If (U={1,2,\ldots,20}), (A={x:x\) is even\(}) and (B={x:x\) is a multiple of \(5}), what does (A'\cap B') represent\)?
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A वे संख्याएँ जो सम भी हैं और (5) के गुणज भी हैं / Numbers that are both even and multiples of (5)
B वे संख्याएँ जो न सम हैं और न (5) के गुणज हैं / Numbers that are neither even nor multiples of (5)
C वे संख्याएँ जो सम हैं पर (5) के गुणज नहीं हैं / Numbers that are even but not multiples of (5)
D वे संख्याएँ जो (5) के गुणज हैं पर सम नहीं हैं / Numbers that are multiples of (5) but not even
Explanation opens after your attempt
Correct Answer
B. वे संख्याएँ जो न सम हैं और न (5) के गुणज हैं / Numbers that are neither even nor multiples of (5)
Step 1
Concept
\(A'\cap B'\) means not in (A) and not in (B). By De Morgan, it is also (\(A\cup B\)').
Step 2
Why this answer is correct
The correct answer is B. वे संख्याएँ जो न सम हैं और न (5) के गुणज हैं / Numbers that are neither even nor multiples of (5). \(A'\cap B'\) means not in (A) and not in (B). By De Morgan, it is also (\(A\cup B\)').
Step 3
Exam Tip
\(A'\cap B'\) का अर्थ है (A) में नहीं और (B) में नहीं। डी मॉर्गन से यह (\(A\cup B\)') भी है।
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यदि (|U|=80), (|A|=35), (|B|=40) और \(|A\cap B|=15\), तो (|\(A\cup B\)'|) कितना है?
If (|U|=80), (|A|=35), (|B|=40) and \(|A\cap B|=15\), what is (|\(A\cup B\)'|)?
#sets
#complement
#cardinality
#union
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A (20)
B (25)
C (30)
D (35)
Explanation opens after your attempt
Step 1
Concept
\(|A\cup B|=35+40-15=60\), so the complement has (80-60=20) elements. Find the union count first.
Step 2
Why this answer is correct
The correct answer is A. (20). \(|A\cup B|=35+40-15=60\), so the complement has (80-60=20) elements. Find the union count first.
Step 3
Exam Tip
\(|A\cup B|=35+40-15=60\), इसलिए पूरक में (80-60=20) अवयव हैं। पहले संघ की संख्या निकालें।
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यदि (|U|=100), (|A'|=58) और (|B'|=47), साथ ही \(|A\cap B|=21\), तो (|\(A\cup B\)'|) कितना होगा?
If (|U|=100), (|A'|=58) and (|B'|=47), with \(|A\cap B|=21\), what is (|\(A\cup B\)'|)?
#sets
#complement
#cardinality
#hard
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A (16)
B (21)
C (26)
D (31)
Explanation opens after your attempt
Step 1
Concept
(|A|=42) and (|B|=53), so \(|A\cup B|=42+53-21=74\). Hence (|\(A\cup B\)'|=100-74=26).
Step 2
Why this answer is correct
The correct answer is C. (26). (|A|=42) and (|B|=53), so \(|A\cup B|=42+53-21=74\). Hence (|\(A\cup B\)'|=100-74=26).
Step 3
Exam Tip
(|A|=42) और (|B|=53), इसलिए \(|A\cup B|=42+53-21=74\)। अतः (|\(A\cup B\)'|=100-74=26)।
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यदि \(A-B=A\cap B'\), तो ((A-B)') किसके बराबर है?
If \(A-B=A\cap B'\), then ((A-B)') is equal to which expression?
#sets
#difference
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#demorgan
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A \(A'\cup B\)
B \(A'\cap B\)
C \(A\cup B'\)
D \(A\cap B\)
Explanation opens after your attempt
Correct Answer
A. \(A'\cup B\)
Step 1
Concept
((A-B)'=\(A\cap B'\)'=A'\cup B). Writing difference using complement is a fast method.
Step 2
Why this answer is correct
The correct answer is A. \(A'\cup B\). ((A-B)'=\(A\cap B'\)'=A'\cup B). Writing difference using complement is a fast method.
Step 3
Exam Tip
((A-B)'=\(A\cap B'\)'=A'\cup B)। अंतर को पूरक के रूप में लिखना तेज तरीका है।
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यदि \(U=\mathbb{R}\) और (A=(2,7]), तो (A') क्या होगा?
If \(U=\mathbb{R}\) and (A=(2,7]), what is (A')?
#sets
#complement
#intervals
#real-numbers
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A \((-\infty,2]\cup\(7,\infty\))
B \(\(-\infty,2\)\cup[7,\infty))
C \(\(-\infty,2\)\cup\(7,\infty\))
D \([2,7))
Explanation opens after your attempt
Correct Answer
A. \((-\infty,2]\cup\(7,\infty\))
Step 1
Concept
(2) was not in the set, so it is included in the complement; (7) was in the set, so it is excluded. Watch open and closed endpoints in intervals.
Step 2
Why this answer is correct
The correct answer is A. \((-\infty,2]\cup\(7,\infty\)). (2) was not in the set, so it is included in the complement; (7) was in the set, so it is excluded. Watch open and closed endpoints in intervals.
Step 3
Exam Tip
(2) समुच्चय में नहीं था, इसलिए पूरक में आएगा; (7) समुच्चय में था, इसलिए पूरक में नहीं आएगा। अंतरालों में खुले और बंद सिरों पर ध्यान दें।
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यदि \(U=\mathbb{R}\), (A=\(-\infty,3\)) और \(B=[1,\infty\)), तो (\(A\cap B\)') क्या होगा?
If \(U=\mathbb{R}\), (A=\(-\infty,3\)) and \(B=[1,\infty\)), what is (\(A\cap B\)')?
#sets
#complement
#intersection
#intervals
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A \(\(-\infty,1\)\cup[3,\infty))
B \((-\infty,1]\cup\(3,\infty\))
C \([1,3))
D \((1,3])
Explanation opens after your attempt
Correct Answer
A. \(\(-\infty,1\)\cup[3,\infty))
Step 1
Concept
\(A\cap B=[1,3\)), so its complement is (\(-\infty,1\)\cup[3,\infty)). First find the correct intersection interval.
Step 2
Why this answer is correct
The correct answer is A. \(\(-\infty,1\)\cup[3,\infty)). \(A\cap B=[1,3\)), so its complement is (\(-\infty,1\)\cup[3,\infty)). First find the correct intersection interval.
Step 3
Exam Tip
\(A\cap B=[1,3\)) है, इसलिए उसका पूरक (\(-\infty,1\)\cup[3,\infty)) होगा। पहले प्रतिच्छेद का सही अंतराल निकालें।
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यदि \(A'\cap B'=\varnothing\), तो \(A\cup B\) के बारे में सही निष्कर्ष क्या है?
If \(A'\cap B'=\varnothing\), what is the correct conclusion about \(A\cup B\)?
#sets
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#reasoning
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A \(A\cup B=\varnothing\)
B \(A\cup B=U\)
C \(A\cap B=U\)
D (A=B')
Explanation opens after your attempt
Correct Answer
B. \(A\cup B=U\)
Step 1
Concept
(A'\cap B'=\(A\cup B\)'), so if it is empty, \(A\cup B=U\). If a complement is empty, the original set is universal.
Step 2
Why this answer is correct
The correct answer is B. \(A\cup B=U\). (A'\cap B'=\(A\cup B\)'), so if it is empty, \(A\cup B=U\). If a complement is empty, the original set is universal.
Step 3
Exam Tip
(A'\cap B'=\(A\cup B\)'), इसलिए यदि यह रिक्त है तो \(A\cup B=U\)। पूरक रिक्त हो तो मूल समुच्चय सार्वत्रिक होता है।
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यदि (\(A\cap B\)'=A'\cap B') किसी (U) में सत्य हो, तो (A) और (B) के बारे में क्या कहा जा सकता है?
If (\(A\cap B\)'=A'\cap B') holds in a universal set (U), what can be said about (A) and (B)?
#sets
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#demorgan
#equality
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A \(A\cup B=A\cap B\)
B \(A\cap B=\varnothing\)
C (A'=B')
D \(A\cup B=U\)
Explanation opens after your attempt
Correct Answer
A. \(A\cup B=A\cap B\)
Step 1
Concept
By De Morgan, (\(A\cap B\)'=A'\cup B'), so \(A'\cup B'=A'\cap B'\). This happens when (A'=B'), hence (A=B).
Step 2
Why this answer is correct
The correct answer is A. \(A\cup B=A\cap B\). By De Morgan, (\(A\cap B\)'=A'\cup B'), so \(A'\cup B'=A'\cap B'\). This happens when (A'=B'), hence (A=B).
Step 3
Exam Tip
डी मॉर्गन से (\(A\cap B\)'=A'\cup B'), इसलिए \(A'\cup B'=A'\cap B'\) होगा। यह तभी होता है जब (A'=B'), अर्थात (A=B)।
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यदि \(U={1,2,\ldots,50}\), (A) (2) के गुणजों का समुच्चय है और (B) (5) के गुणजों का समुच्चय है, तो \(|A'\cap B'|\) कितना है?
If \(U={1,2,\ldots,50}\), (A) is the set of multiples of (2) and (B) is the set of multiples of (5), what is \(|A'\cap B'|\)?
#sets
#complement
#counting
#multiples
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A (20)
B (25)
C (30)
D (35)
Explanation opens after your attempt
Step 1
Concept
\(|A\cup B|=25+10-5=30\), so (|A'\cap B'|=|\(A\cup B\)'|=50-30=20). Connect De Morgan with counting.
Step 2
Why this answer is correct
The correct answer is A. (20). \(|A\cup B|=25+10-5=30\), so (|A'\cap B'|=|\(A\cup B\)'|=50-30=20). Connect De Morgan with counting.
Step 3
Exam Tip
\(|A\cup B|=25+10-5=30\), इसलिए (|A'\cap B'|=|\(A\cup B\)'|=50-30=20)। डी मॉर्गन को गिनती से जोड़ें।
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यदि \(U=\{a,b,c,d,e,f,g,h\}\), (A'={b,d,f,h}) और \(B=\{a,b,c,d\}\), तो \(A\cap B'\) क्या है?
If \(U=\{a,b,c,d,e,f,g,h\}\), (A'={b,d,f,h}) and \(B=\{a,b,c,d\}\), what is \(A\cap B'\)?
#sets
#complement
#intersection
#finite-set
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A \({e,g})
B \({a,c})
C \({f,h})
D \({b,d})
Explanation opens after your attempt
Correct Answer
A. \({e,g})
Step 1
Concept
(A=U-A'={a,c,e,g}) and (B'={e,f,g,h}), so \(A\cap B'={e,g}\). First recover the original set from its complement.
Step 2
Why this answer is correct
The correct answer is A. \({e,g}). (A=U-A'={a,c,e,g}) and (B'={e,f,g,h}), so \(A\cap B'={e,g}\). First recover the original set from its complement.
Step 3
Exam Tip
(A=U-A'={a,c,e,g}) और (B'={e,f,g,h}), इसलिए \(A\cap B'={e,g}\)। दिए गए पूरक से पहले मूल समुच्चय निकालें।
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यदि \(A\cup A'=U\) और \(A\cap A'=\varnothing\), तो (A') को कौन सा समुच्चय बनाता है?
If \(A\cup A'=U\) and \(A\cap A'=\varnothing\), what kind of set is (A') with respect to (A)?
#sets
#complement
#basic-property
#identity
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A (A) का समान समुच्चय / Same set as (A)
B (A) का उपसमुच्चय / Subset of (A)
C (A) का पूरक समुच्चय / Complement of (A)
D (A) का संघ समुच्चय / Union set of (A)
Explanation opens after your attempt
Correct Answer
C. (A) का पूरक समुच्चय / Complement of (A)
Step 1
Concept
Both conditions show that (A) and (A') together form (U) and are disjoint. This is the main identity of complement.
Step 2
Why this answer is correct
The correct answer is C. (A) का पूरक समुच्चय / Complement of (A). Both conditions show that (A) and (A') together form (U) and are disjoint. This is the main identity of complement.
Step 3
Exam Tip
दोनों शर्तें बताती हैं कि (A) और (A') मिलकर (U) बनाते हैं और अलग-अलग हैं। यही पूरक की मुख्य पहचान है।
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\(यदि (U={x:x\in \mathbb{N},x\le 40}), (A={x:x\) 4 से विभाज्य है\(}) और (B={x:x\) 6 से विभाज्य है\(}), तो (|(A\cap B)'|) कितना है\)?
\(If (U={x:x\in \mathbb{N},x\le 40}), (A={x:x\) is divisible by \(4}) and (B={x:x\) is divisible by \(6}), what is (|(A\cap B)'|)\)?
#sets
#complement
#lcm
#counting
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A (36)
B (37)
C (38)
D (39)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B\) contains multiples of (12), namely (12,24,36), so it has (3) elements. Hence the complement has (40-3=37) elements.
Step 2
Why this answer is correct
The correct answer is B. (37). \(A\cap B\) contains multiples of (12), namely (12,24,36), so it has (3) elements. Hence the complement has (40-3=37) elements.
Step 3
Exam Tip
\(A\cap B\) में (12) के गुणज (12,24,36) हैं, इसलिए (3) अवयव हैं। अतः पूरक में (40-3=37) अवयव होंगे।
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यदि \(U=\mathbb{R}\), \(A={x:x^2-5x+6=0}\), तो (A') का सही वर्णन कौन सा है?
If \(U=\mathbb{R}\), \(A={x:x^2-5x+6=0}\), which is the correct description of (A')?
#sets
#complement
#quadratic
#set-builder
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A \(\mathbb{R}-{2,3})
B \({2,3})
C \(\mathbb{R}-{1,6})
D \(\varnothing)
Explanation opens after your attempt
Correct Answer
A. \(\mathbb{R}-{2,3})
Step 1
Concept
The equation gives \(A=\{2,3\}\), so the complement is \(\mathbb{R}-{2,3}\). Solve the equation first.
Step 2
Why this answer is correct
The correct answer is A. \(\mathbb{R}-{2,3}). The equation gives (A={2,3}), so the complement is \(\mathbb{R}-{2,3}\). Solve the equation first.
Step 3
Exam Tip
समीकरण से \(A=\{2,3\}\) मिलता है, इसलिए पूरक \(\mathbb{R}-{2,3}\) होगा। पहले समीकरण का हल निकालें।
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यदि \(U={1,2,\ldots,12}\), \(A=\{1,3,5,7,9,11\}\) और \(B=\{2,3,5,7,11\}\), तो (\(A'\cup B'\)') क्या है?
If \(U={1,2,\ldots,12}\), \(A=\{1,3,5,7,9,11\}\) and \(B=\{2,3,5,7,11\}\), what is (\(A'\cup B'\)')?
#sets
#demorgan
#complement
#prime-odd
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A \({3,5,7,11})
B \({1,9})
C \({2,4,6,8,10,12})
D \({1,2,9})
Explanation opens after your attempt
Correct Answer
A. \({3,5,7,11})
Step 1
Concept
By De Morgan, (\(A'\cup B'\)'=A\cap B), which is ({3,5,7,11}). Simplify the outer complement using the law.
Step 2
Why this answer is correct
The correct answer is A. \({3,5,7,11}). By De Morgan, (\(A'\cup B'\)'=A\cap B), which is ({3,5,7,11}). Simplify the outer complement using the law.
Step 3
Exam Tip
डी मॉर्गन से (\(A'\cup B'\)'=A\cap B), जो ({3,5,7,11}) है। बाहरी पूरक को नियम से सरल करें।
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यदि (A'=B'), तो (A) और (B) के लिए सही निष्कर्ष क्या है?
If (A'=B'), what is the correct conclusion for (A) and (B)?
#sets
#complement
#double-complement
#equality
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A \(A\cap B=\varnothing\)
B \(A\cup B=U\)
C (A=B)
D (A=B')
Explanation opens after your attempt
Step 1
Concept
Taking complement on both sides gives ((A')'=(B')'), so (A=B). The double complement gives the original set.
Step 2
Why this answer is correct
The correct answer is C. (A=B). Taking complement on both sides gives ((A')'=(B')'), so (A=B). The double complement gives the original set.
Step 3
Exam Tip
दोनों पक्षों का पूरक लेने पर ((A')'=(B')'), इसलिए (A=B)। दोहरा पूरक मूल समुच्चय देता है।
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यदि \(U={1,2,\ldots,60}\), (A) (3) के गुणजों का समुच्चय है, (B) (4) के गुणजों का समुच्चय है और (C) (5) के गुणजों का समुच्चय है, तो (|\(A\cup B\cup C\)'|) कितना है?
If \(U={1,2,\ldots,60}\), (A) is the set of multiples of (3), (B) is the set of multiples of (4), and (C) is the set of multiples of (5), what is (|\(A\cup B\cup C\)'|)?
#sets
#complement
#three-sets
#inclusion-exclusion
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A (20)
B (22)
C (24)
D (26)
Explanation opens after your attempt
Step 1
Concept
By inclusion-exclusion, \(|A\cup B\cup C|=20+15+12-5-4-3+1=36\). So the complement has (60-36=24) elements.
Step 2
Why this answer is correct
The correct answer is C. (24). By inclusion-exclusion, \(|A\cup B\cup C|=20+15+12-5-4-3+1=36\). So the complement has (60-36=24) elements.
Step 3
Exam Tip
समावेशन-बहिष्करण से \(|A\cup B\cup C|=20+15+12-5-4-3+1=36\)। इसलिए पूरक में (60-36=24) अवयव हैं।
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यदि (U) में (120) विद्यार्थी हैं, (70) गणित पढ़ते हैं, (65) भौतिकी पढ़ते हैं और (30) दोनों पढ़ते हैं, तो न गणित न भौतिकी पढ़ने वालों की संख्या कितनी है?
In a universal set (U) of (120) students, (70) study mathematics, (65) study physics, and (30) study both. How many study neither mathematics nor physics?
#sets
#complement
#word-problem
#cardinality
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A (10)
B (15)
C (20)
D (25)
Explanation opens after your attempt
Step 1
Concept
\(|M\cup P|=70+65-30=105\), so (|\(M\cup P\)'|=120-105=15). Neither means the complement of the union.
Step 2
Why this answer is correct
The correct answer is B. (15). \(|M\cup P|=70+65-30=105\), so (|\(M\cup P\)'|=120-105=15). Neither means the complement of the union.
Step 3
Exam Tip
\(|M\cup P|=70+65-30=105\), इसलिए (|\(M\cup P\)'|=120-105=15)। न यह न वह का अर्थ पूरक होता है।
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\(यदि (U={x:x\in \mathbb{Z},0\le x\le 15}), (A={x:x\) अभाज्य है}), तो (A') में कौन सा अवयव अवश्य होगा?
\(If (U={x:x\in \mathbb{Z},0\le x\le 15}), (A={x:x\) is prime}), which element must be in (A')?
#sets
#complement
#prime
#integers
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A (2)
B (3)
C (9)
D (13)
Explanation opens after your attempt
Step 1
Concept
(9) is not prime and belongs to (U), so it is in (A'). Complement contains elements in (U) but not in (A).
Step 2
Why this answer is correct
The correct answer is C. (9). (9) is not prime and belongs to (U), so it is in (A'). Complement contains elements in (U) but not in (A).
Step 3
Exam Tip
(9) अभाज्य नहीं है और (U) में है, इसलिए (A') में होगा। पूरक में वे अवयव आते हैं जो (U) में हों पर (A) में न हों।
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यदि \(U={1,2,\ldots,10}\), \(A=\{1,2,3,4\}\) और \(B=\{3,4,5,6\}\), तो \(A'\Delta B'\) किसके बराबर है?
If \(U={1,2,\ldots,10}\), \(A=\{1,2,3,4\}\) and \(B=\{3,4,5,6\}\), what is \(A'\Delta B'\) equal to?
#sets
#complement
#symmetric-difference
#property
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A \(A\Delta B\)
B \(A\cap B\)
C \(A'\cap B'\)
D (U)
Explanation opens after your attempt
Correct Answer
A. \(A\Delta B\)
Step 1
Concept
The symmetric difference of complements equals the symmetric difference of the original sets. Here both give ({1,2,5,6}).
Step 2
Why this answer is correct
The correct answer is A. \(A\Delta B\). The symmetric difference of complements equals the symmetric difference of the original sets. Here both give ({1,2,5,6}).
Step 3
Exam Tip
दो समुच्चयों के पूरकों का सममित अंतर मूल समुच्चयों के सममित अंतर के बराबर होता है। यहां दोनों में केवल ({1,2,5,6}) मिलेंगे।
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\(यदि (U=\mathbb{R}) और (A={x:x<2\) या \(x\ge 8}), तो (A') क्या है\)?
\(If (U=\mathbb{R}) and (A={x:x<2\) or \(x\ge 8}), what is (A')\)?
#sets
#complement
#inequality
#intervals
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A \([2,8))
B \((2,8])
C \(\(-\infty,2\)\cup[8,\infty))
D \((-\infty,2]\cup\(8,\infty\))
Explanation opens after your attempt
Correct Answer
A. \([2,8))
Step 1
Concept
Outside (A) are the real numbers satisfying \(2\le x<8\). Therefore (A'=[2,8)).
Step 2
Why this answer is correct
The correct answer is A. \([2,8)). Outside (A) are the real numbers satisfying \(2\le x<8\). Therefore (A'=[2,8)).
Step 3
Exam Tip
(A) के बाहर वे वास्तविक संख्याएँ हैं जिनके लिए \(2\le x<8\)। इसलिए (A'=[2,8)) है।
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यदि \(U=\mathbb{R}\), \(A={x:x^2<16}\), तो (A') क्या होगा?
If \(U=\mathbb{R}\), \(A={x:x^2<16}\), what is (A')?
#sets
#complement
#quadratic-inequality
#real-numbers
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A \(\(-\infty,-4]\cup[4,\infty\))
B \(\(-\infty,-4\)\cup\(4,\infty\))
C \((-4,4))
D \([-4,4])
Explanation opens after your attempt
Correct Answer
A. \(\(-\infty,-4]\cup[4,\infty\))
Step 1
Concept
\(x^2<16\Rightarrow -4<x<4\), so the complement has \(x\le -4\) or \(x\ge 4\). A strict inequality adds equality in the complement.
Step 2
Why this answer is correct
The correct answer is A. \(\(-\infty,-4]\cup[4,\infty\)). \(x^2<16\Rightarrow -4<x<4\), so the complement has \(x\le -4\) or \(x\ge 4\). A strict inequality adds equality in the complement.
Step 3
Exam Tip
\(x^2<16\Rightarrow -4<x<4\), इसलिए पूरक में \(x\le -4\) या \(x\ge 4\) होगा। सख्त असमानता पूरक में बराबरी जोड़ देती है।
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यदि \(A\cap B=A\cap C\) और \(A'\cap B=A'\cap C\), तो (B) और (C) के बारे में सही निष्कर्ष क्या है?
If \(A\cap B=A\cap C\) and \(A'\cap B=A'\cap C\), what is the correct conclusion about (B) and (C)?
#sets
#complement
#partition
#reasoning
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A (B=C)
B (B=C')
C \(B\cap C=\varnothing\)
D \(B\cup C=U\)
Explanation opens after your attempt
Step 1
Concept
(B=\(A\cap B\)\cup\(A'\cap B\)) and (C=\(A\cap C\)\cup\(A'\cap C\)), so (B=C). (A) and (A') partition (U).
Step 2
Why this answer is correct
The correct answer is A. (B=C). (B=\(A\cap B\)\cup\(A'\cap B\)) and (C=\(A\cap C\)\cup\(A'\cap C\)), so (B=C). (A) and (A') partition (U).
Step 3
Exam Tip
(B=\(A\cap B\)\cup\(A'\cap B\)) और (C=\(A\cap C\)\cup\(A'\cap C\)), इसलिए (B=C)। (A) और (A') पूरे (U) को बांटते हैं।
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\(यदि (U={1,2,\ldots,25}), (A={x:x\) विषम है\(}) और (B={x:x\) पूर्ण वर्ग है\(}), तो (|A'\cap B|) कितना है\)?
\(If (U={1,2,\ldots,25}), (A={x:x\) is odd\(}) and (B={x:x\) is a perfect square\(}), what is (|A'\cap B|)\)?
#sets
#complement
#perfect-square
#intersection
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
(A') is the set of even numbers and \(B=\{1,4,9,16,25\}\). Their intersection is ({4,16}), so the count is (2).
Step 2
Why this answer is correct
The correct answer is B. (2). (A') is the set of even numbers and \(B=\{1,4,9,16,25\}\). Their intersection is ({4,16}), so the count is (2).
Step 3
Exam Tip
(A') सम संख्याएँ हैं और \(B=\{1,4,9,16,25\}\)। इनके प्रतिच्छेद में ({4,16}) हैं, इसलिए संख्या (2) है।
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\(यदि (U={x:x\in \mathbb{N},x\le 100}), (A={x:x\) 2 से विभाज्य नहीं है}), तो (A') किसका समुच्चय है?
\(If (U={x:x\in \mathbb{N},x\le 100}), (A={x:x\) is not divisible by 2}), then (A') is the set of what?
#sets
#complement
#divisibility
#logic
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A (2) से विभाज्य संख्याएँ / Numbers divisible by (2)
B (2) से विभाज्य नहीं संख्याएँ / Numbers not divisible by (2)
C सभी अभाज्य संख्याएँ / All prime numbers
D सभी पूर्ण वर्ग / All perfect squares
Explanation opens after your attempt
Correct Answer
A. (2) से विभाज्य संख्याएँ / Numbers divisible by (2)
Step 1
Concept
(A) contains numbers not divisible by (2), so its complement contains numbers divisible by (2). Be careful with complements of negative descriptions.
Step 2
Why this answer is correct
The correct answer is A. (2) से विभाज्य संख्याएँ / Numbers divisible by (2). (A) contains numbers not divisible by (2), so its complement contains numbers divisible by (2). Be careful with complements of negative descriptions.
Step 3
Exam Tip
(A) में (2) से विभाज्य नहीं संख्याएँ हैं, इसलिए पूरक में (2) से विभाज्य संख्याएँ होंगी। नकारात्मक परिभाषा में पूरक लेते समय सावधान रहें।
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यदि (\(A\cup B\)'=\varnothing) और \(A\cap B=\varnothing\), तो (B) किसके बराबर होगा?
If (\(A\cup B\)'=\varnothing) and \(A\cap B=\varnothing\), then (B) is equal to what?
#sets
#complement
#disjoint
#partition
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A (A)
B (A')
C (U)
D \(\varnothing)
Explanation opens after your attempt
Step 1
Concept
The first condition gives \(A\cup B=U\), and the second says they are disjoint. Therefore (B) is exactly (A').
Step 2
Why this answer is correct
The correct answer is B. (A'). The first condition gives \(A\cup B=U\), and the second says they are disjoint. Therefore (B) is exactly (A').
Step 3
Exam Tip
पहली शर्त से \(A\cup B=U\) और दूसरी से दोनों असंयुक्त हैं। इसलिए (B) ठीक (A') है।
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यदि \(U={1,2,\ldots,18}\), \(A=\{2,4,6,8,10,12,14,16,18\}\), तो ((A')') क्या है?
If \(U={1,2,\ldots,18}\), \(A=\{2,4,6,8,10,12,14,16,18\}\), what is ((A')')?
#sets
#complement
#double-complement
#identity
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A (A)
B (A')
C (U)
D \(\varnothing)
Explanation opens after your attempt
Step 1
Concept
The double complement gives the original set, so ((A')'=A). This identity is useful in many complement questions.
Step 2
Why this answer is correct
The correct answer is A. (A). The double complement gives the original set, so ((A')'=A). This identity is useful in many complement questions.
Step 3
Exam Tip
दोहरा पूरक मूल समुच्चय देता है, इसलिए ((A')'=A)। यह पहचान लगभग हर पूरक प्रश्न में काम आती है।
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\(यदि (U={1,2,\ldots,20}), (A={x:x\) 3 से विभाज्य है}), तो (A') में अवयवों की संख्या कितनी है?
\(If (U={1,2,\ldots,20}), (A={x:x\) is divisible by 3}), how many elements are in (A')?
#sets
#complement
#counting
#multiples
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A (13)
B (14)
C (15)
D (16)
Explanation opens after your attempt
Step 1
Concept
There are (6) multiples of (3) up to (20), so (A') has (20-6=14) elements. For complement count, subtract from the total.
Step 2
Why this answer is correct
The correct answer is B. (14). There are (6) multiples of (3) up to (20), so (A') has (20-6=14) elements. For complement count, subtract from the total.
Step 3
Exam Tip
(20) तक (3) के (6) गुणज हैं, इसलिए (A') में (20-6=14) अवयव हैं। पूरक की संख्या के लिए कुल में से घटाएं।
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यदि \(U={x:x\in \mathbb{Z},-3\le x\le 6}\) और \(A={x:x\ge 2}\), तो (A') क्या होगा?
If \(U={x:x\in \mathbb{Z},-3\le x\le 6}\) and \(A={x:x\ge 2}\), what is (A')?
#sets
#complement
#integers
#inequality
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A \({-3,-2,-1,0,1})
B \({2,3,4,5,6})
C \({-3,-2,-1,0})
D \({1,2,3,4,5,6})
Explanation opens after your attempt
Correct Answer
A. \({-3,-2,-1,0,1})
Step 1
Concept
(A) contains integers from (2) to (6), so (A') contains integers of (U) less than (2). Do not ignore the universal boundary.
Step 2
Why this answer is correct
The correct answer is A. \({-3,-2,-1,0,1}). (A) contains integers from (2) to (6), so (A') contains integers of (U) less than (2). Do not ignore the universal boundary.
Step 3
Exam Tip
(A) में (2) से (6) तक के पूर्णांक हैं, इसलिए (A') में (2) से छोटे (U) के पूर्णांक होंगे। सार्वत्रिक सीमा को न भूलें।
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\(यदि (U={1,2,\ldots,30}), (A={x:x\) 2 या 3 से विभाज्य है}), तो (A') में कितने अवयव हैं?
\(If (U={1,2,\ldots,30}), (A={x:x\) is divisible by 2 or 3}), how many elements are in (A')?
#sets
#complement
#inclusion-exclusion
#divisibility
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
Numbers divisible by (2) or (3) are (15+10-5=20). Therefore the complement contains (30-20=10) numbers.
Step 2
Why this answer is correct
The correct answer is C. (10). Numbers divisible by (2) or (3) are (15+10-5=20). Therefore the complement contains (30-20=10) numbers.
Step 3
Exam Tip
(2) या (3) से विभाज्य संख्याएँ (15+10-5=20) हैं। इसलिए पूरक में (30-20=10) संख्याएँ हैं।
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यदि \(U={1,2,\ldots,16}\), \(A=\{1,2,4,8,16\}\) और \(B=\{2,4,6,8,10,12,14,16\}\), तो (B-A) किसके बराबर है?
If \(U={1,2,\ldots,16}\), \(A=\{1,2,4,8,16\}\) and \(B=\{2,4,6,8,10,12,14,16\}\), what is (B-A) equal to?
#sets
#complement
#difference
#identity
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A \(B\cap A'\)
B \(B\cup A'\)
C \(A\cap B'\)
D \(A'\cap B'\)
Explanation opens after your attempt
Correct Answer
A. \(B\cap A'\)
Step 1
Concept
The difference (B-A) means in (B) and not in (A), that is \(B\cap A'\). Writing difference with complement is easier.
Step 2
Why this answer is correct
The correct answer is A. \(B\cap A'\). The difference (B-A) means in (B) and not in (A), that is \(B\cap A'\). Writing difference with complement is easier.
Step 3
Exam Tip
समुच्चय अंतर (B-A) का अर्थ है (B) में और (A) में नहीं, यानी \(B\cap A'\)। अंतर को पूरक से लिखना आसान है।
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यदि \(U=\mathbb{R}\), (A=[-2,5)) और (B=(0,7]), तो (\(A\cup B\)') क्या होगा?
If \(U=\mathbb{R}\), (A=[-2,5)) and (B=(0,7]), what is (\(A\cup B\)')?
#sets
#complement
#union
#intervals
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A \(\(-\infty,-2\)\cup\(7,\infty\))
B \(\(-\infty,-2]\cup[7,\infty\))
C \([-2,7])
D \(\(-\infty,0]\cup[5,\infty\))
Explanation opens after your attempt
Correct Answer
A. \(\(-\infty,-2\)\cup\(7,\infty\))
Step 1
Concept
\(A\cup B=[-2,7]\), so the complement is (\(-\infty,-2\)\cup\(7,\infty\)). First form the union interval correctly.
Step 2
Why this answer is correct
The correct answer is A. \(\(-\infty,-2\)\cup\(7,\infty\)). \(A\cup B=[-2,7]\), so the complement is (\(-\infty,-2\)\cup\(7,\infty\)). First form the union interval correctly.
Step 3
Exam Tip
\(A\cup B=[-2,7]\), इसलिए पूरक (\(-\infty,-2\)\cup\(7,\infty\)) है। पहले संघ का अंतराल सही बनाएं।
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यदि \(U=\mathbb{R}\), (A=(-3,4]) और (B=[1,6)), तो (\(A'\cap B'\)) क्या होगा?
If \(U=\mathbb{R}\), (A=(-3,4]) and (B=[1,6)), what is \(A'\cap B'\)?
#sets
#complement
#demorgan
#intervals
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A \(\(-\infty,-3]\cup[6,\infty\))
B \(\(-\infty,-3\)\cup\(6,\infty\))
C \((-3,6))
D \([1,4])
Explanation opens after your attempt
Correct Answer
A. \(\(-\infty,-3]\cup[6,\infty\))
Step 1
Concept
(A'\cap B'=\(A\cup B\)') and \(A\cup B=(-3,6)\). So the complement is (\(-\infty,-3]\cup[6,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. \(\(-\infty,-3]\cup[6,\infty\)). (A'\cap B'=\(A\cup B\)') and \(A\cup B=(-3,6)\). So the complement is (\(-\infty,-3]\cup[6,\infty\)).
Step 3
Exam Tip
(A'\cap B'=\(A\cup B\)') और \(A\cup B=(-3,6)\)। इसलिए पूरक (\(-\infty,-3]\cup[6,\infty\)) है।
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\(यदि (U={1,2,\ldots,100}), (A={x:x\) पूर्ण वर्ग है}), तो (|A'|) कितना है?
\(If (U={1,2,\ldots,100}), (A={x:x\) is a perfect square}), what is (|A'|)?
#sets
#complement
#perfect-square
#counting
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A (89)
B (90)
C (91)
D (92)
Explanation opens after your attempt
Step 1
Concept
Perfect squares up to (100) are from \(1^2\) to \(10^2\), giving (10) elements. Hence (|A'|=100-10=90).
Step 2
Why this answer is correct
The correct answer is B. (90). Perfect squares up to (100) are from \(1^2\) to \(10^2\), giving (10) elements. Hence (|A'|=100-10=90).
Step 3
Exam Tip
(100) तक पूर्ण वर्ग \(1^2\) से \(10^2\) तक (10) हैं। इसलिए (|A'|=100-10=90)।
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यदि \(U={1,2,\ldots,15}\), \(A=\{1,4,9\}\), \(B=\{2,4,6,8,10,12,14\}\), तो (\(A\cap B\)') में कितने अवयव हैं?
If \(U={1,2,\ldots,15}\), \(A=\{1,4,9\}\), \(B=\{2,4,6,8,10,12,14\}\), how many elements are in (\(A\cap B\)')?
#sets
#complement
#intersection
#counting
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A (13)
B (14)
C (15)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B={4}\), so its complement has (15-1=14) elements. The complement of a small intersection can be large.
Step 2
Why this answer is correct
The correct answer is B. (14). \(A\cap B={4}\), so its complement has (15-1=14) elements. The complement of a small intersection can be large.
Step 3
Exam Tip
\(A\cap B={4}\), इसलिए उसके पूरक में (15-1=14) अवयव हैं। छोटे प्रतिच्छेद का पूरक बड़ा हो सकता है।
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यदि (U) में (n) अवयव हैं और (A) में (r) अवयव हैं, तो (A') में कितने अवयव होंगे?
If (U) has (n) elements and (A) has (r) elements, how many elements are in (A')?
#sets
#complement
#formula
#cardinality
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A (n+r)
B (n-r)
C (r-n)
D (nr)
Explanation opens after your attempt
Step 1
Concept
The complement contains elements of (U) not in (A), so the number is (n-r). Use this formula only when \(A\subseteq U\).
Step 2
Why this answer is correct
The correct answer is B. (n-r). The complement contains elements of (U) not in (A), so the number is (n-r). Use this formula only when \(A\subseteq U\).
Step 3
Exam Tip
पूरक में (U) के वे अवयव हैं जो (A) में नहीं हैं, इसलिए संख्या (n-r) है। यह सूत्र तभी लागू करें जब \(A\subseteq U\) हो।
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यदि \(A\subseteq U\), तो \(A\cup A'\) और \(A\cap A'\) क्रमशः क्या हैं?
If \(A\subseteq U\), what are \(A\cup A'\) and \(A\cap A'\) respectively?
#sets
#complement
#identity
#union-intersection
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A \(U,\varnothing\)
B \(\varnothing,U)
C (A,U)
D (A',A)
Explanation opens after your attempt
Correct Answer
A. \(U,\varnothing\)
Step 1
Concept
(A) and (A') together make the whole (U), and their common part is empty. This is the paired identity of complement.
Step 2
Why this answer is correct
The correct answer is A. \(U,\varnothing\). (A) and (A') together make the whole (U), and their common part is empty. This is the paired identity of complement.
Step 3
Exam Tip
(A) और (A') मिलकर पूरा (U) बनाते हैं तथा उनका साझा भाग रिक्त होता है। यह पूरक की जोड़ी पहचान है।
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\(यदि (U={1,2,\ldots,24}), (A={x:x\) 2 से विभाज्य है\(}), (B={x:x\) 3 से विभाज्य है\(}), तो (|A'\cup B'|) कितना है\)?
\(If (U={1,2,\ldots,24}), (A={x:x\) is divisible by \(2}), (B={x:x\) is divisible by \(3}), what is (|A'\cup B'|)\)?
#sets
#complement
#demorgan
#multiples
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A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
(A'\cup B'=\(A\cap B\)'). \(A\cap B\) has (4) multiples of (6), so the count is (24-4=20).
Step 2
Why this answer is correct
The correct answer is C. (20). (A'\cup B'=\(A\cap B\)'). \(A\cap B\) has (4) multiples of (6), so the count is (24-4=20).
Step 3
Exam Tip
(A'\cup B'=\(A\cap B\)')। \(A\cap B\) में (6) के (4) गुणज हैं, इसलिए संख्या (24-4=20) है।
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यदि \(U={1,2,\ldots,12}\) और (A'={1,5,7,11}), तो (A) क्या है?
If \(U={1,2,\ldots,12}\) and (A'={1,5,7,11}), what is (A)?
#sets
#complement
#finite-set
#reconstruction
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A \({2,3,4,6,8,9,10,12})
B \({1,5,7,11})
C \({2,4,6,8,10,12})
D \({3,6,9,12})
Explanation opens after your attempt
Correct Answer
A. \({2,3,4,6,8,9,10,12})
Step 1
Concept
(A) contains the elements of (U) that are not in (A'). Therefore \(A=\{2,3,4,6,8,9,10,12\}\).
Step 2
Why this answer is correct
The correct answer is A. \({2,3,4,6,8,9,10,12}). (A) contains the elements of (U) that are not in (A'). Therefore \(A=\{2,3,4,6,8,9,10,12\}\).
Step 3
Exam Tip
(A) में (U) के वे अवयव होंगे जो (A') में नहीं हैं। इसलिए \(A=\{2,3,4,6,8,9,10,12\}\) है।
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\(यदि (U=\mathbb{R}), (A={x:x\le -1\) या \(x>4}), तो (A') क्या है\)?
\(If (U=\mathbb{R}), (A={x:x\le -1\) or \(x>4}), what is (A')\)?
#sets
#complement
#inequality
#interval
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A \((-1,4])
B \([-1,4))
C \((-1,4))
D \([-1,4])
Explanation opens after your attempt
Correct Answer
A. \((-1,4])
Step 1
Concept
To be outside (A), (x>-1) and \(x\le 4\) must hold. Therefore (A'=(-1,4]).
Step 2
Why this answer is correct
The correct answer is A. \((-1,4]). To be outside (A), (x>-1) and \(x\le 4\) must hold. Therefore (A'=(-1,4]).
Step 3
Exam Tip
(A) के बाहर रहने के लिए (x>-1) और \(x\le 4\) होना चाहिए। इसलिए (A'=(-1,4]) है।
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यदि (U) में (200) विद्यार्थी हैं, (120) हिंदी लेते हैं, (90) अंग्रेजी लेते हैं और (50) दोनों लेते हैं, तो केवल कोई भी भाषा न लेने वालों की संख्या क्या है?
If (U) has (200) students, (120) take Hindi, (90) take English, and (50) take both, how many take neither language?
#sets
#complement
#word-problem
#students
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A (40)
B (45)
C (50)
D (55)
Explanation opens after your attempt
Step 1
Concept
\(|H\cup E|=120+90-50=160\), so (|\(H\cup E\)'|=200-160=40). Read neither as (\(H\cup E\)').
Step 2
Why this answer is correct
The correct answer is A. (40). \(|H\cup E|=120+90-50=160\), so (|\(H\cup E\)'|=200-160=40). Read neither as (\(H\cup E\)').
Step 3
Exam Tip
\(|H\cup E|=120+90-50=160\), इसलिए (|\(H\cup E\)'|=200-160=40)। neither को (\(H\cup E\)') के रूप में पढ़ें।
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यदि \(A\cap B'=\varnothing\), तो कौन सा निष्कर्ष सदैव सत्य है?
If \(A\cap B'=\varnothing\), which conclusion is always true?
#sets
#complement
#subset
#reasoning
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A \(A\subseteq B\)
B \(B\subseteq A\)
C (A=B')
D \(A\cup B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \(A\subseteq B\)
Step 1
Concept
\(A\cap B'=\varnothing\) means no element of (A) lies outside (B). Hence every element of (A) is in (B).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq B\). \(A\cap B'=\varnothing\) means no element of (A) lies outside (B). Hence every element of (A) is in (B).
Step 3
Exam Tip
\(A\cap B'=\varnothing\) का अर्थ है (A) का कोई अवयव (B) के बाहर नहीं है। इसलिए हर (A) का अवयव (B) में है।
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यदि \(A\cup B'=U\), तो कौन सा संबंध सदैव सत्य है?
If \(A\cup B'=U\), which relation is always true?
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#complement
#subset
#logic
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A \(B\subseteq A\)
B \(A\subseteq B\)
C (A=B')
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \(B\subseteq A\)
Step 1
Concept
Taking complement of \(A\cup B'=U\) gives \(A'\cap B=\varnothing\), so \(B\subseteq A\). Taking complements makes the relation easier.
Step 2
Why this answer is correct
The correct answer is A. \(B\subseteq A\). Taking complement of \(A\cup B'=U\) gives \(A'\cap B=\varnothing\), so \(B\subseteq A\). Taking complements makes the relation easier.
Step 3
Exam Tip
\(A\cup B'=U\) का पूरक लेने पर \(A'\cap B=\varnothing\), इसलिए \(B\subseteq A\)। पूरक लेकर संबंध आसान हो जाता है।
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यदि \(U={1,2,\ldots,36}\), (A) (4) के गुणजों का समुच्चय है और (B) (9) के गुणजों का समुच्चय है, तो \(|A'\cap B'|\) कितना है?
If \(U={1,2,\ldots,36}\), (A) is the set of multiples of (4) and (B) is the set of multiples of (9), what is \(|A'\cap B'|\)?
#sets
#complement
#counting
#demorgan
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A (23)
B (24)
C (25)
D (26)
Explanation opens after your attempt
Step 1
Concept
(|A|=9), (|B|=4), and \(|A\cap B|=1\), so \(|A\cup B|=12\). Hence \(|A'\cap B'|=36-12=24\).
Step 2
Why this answer is correct
The correct answer is B. (24). (|A|=9), (|B|=4), and \(|A\cap B|=1\), so \(|A\cup B|=12\). Hence \(|A'\cap B'|=36-12=24\).
Step 3
Exam Tip
(|A|=9), (|B|=4) और \(|A\cap B|=1\), इसलिए \(|A\cup B|=12\)। अतः \(|A'\cap B'|=36-12=24\)।
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यदि \(U=\mathbb{R}\), \(A={x:x^2-1\ge 0}\), तो (A') क्या होगा?
If \(U=\mathbb{R}\), \(A={x:x^2-1\ge 0}\), what is (A')?
#sets
#complement
#quadratic-inequality
#real-numbers
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A \((-1,1))
B \([-1,1])
C \(\(-\infty,-1]\cup[1,\infty\))
D \(\(-\infty,-1\)\cup\(1,\infty\))
Explanation opens after your attempt
Correct Answer
A. \((-1,1))
Step 1
Concept
The solution of \(x^2-1\ge 0\) is \(x\le -1\) or \(x\ge 1\). Its complement is (-1<x<1), that is ((-1,1)).
Step 2
Why this answer is correct
The correct answer is A. \((-1,1)). The solution of \(x^2-1\ge 0\) is \(x\le -1\) or \(x\ge 1\). Its complement is (-1<x<1), that is ((-1,1)).
Step 3
Exam Tip
\(x^2-1\ge 0\) का हल \(x\le -1\) या \(x\ge 1\) है। इसका पूरक (-1<x<1), यानी ((-1,1)) है।
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