(A) has (16) elements, (B) has (12), and \(A\cap B\) has (4) multiples of (24), so \(|A\cup B|=24\). Hence (|\(A\cup B\)'|=96-24=72).
Step 2
Why this answer is correct
The correct answer is C. (72). (A) has (16) elements, (B) has (12), and \(A\cap B\) has (4) multiples of (24), so \(|A\cup B|=24\). Hence (|\(A\cup B\)'|=96-24=72).
Step 3
Exam Tip
(A) में (16), (B) में (12) और \(A\cap B\) में (24) के (4) गुणज हैं, इसलिए \(|A\cup B|=24\)। अतः (|\(A\cup B\)'|=96-24=72)।
A. ({-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,7,8,9,10})
Step 1
Concept
\(x^2-9x+20\le 0\Rightarrow 4\le x\le 5\), so \(A=\{4,5\}\). The complement is obtained by removing (4,5) from (U).
Step 2
Why this answer is correct
The correct answer is A. ({-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,7,8,9,10}). \(x^2-9x+20\le 0\Rightarrow 4\le x\le 5\), so \(A=\{4,5\}\). The complement is obtained by removing (4,5) from (U).
Step 3
Exam Tip
\(x^2-9x+20\le 0\Rightarrow 4\le x\le 5\) नहीं, सही हल \(4\le x\le 5\) है, इसलिए \(A=\{4,5\}\)। दिए गए विकल्पों में (A') वही है जो (U) से (4,5) हटाकर बनता है।
\(A\cup B=(-6,5)\), because (-6) and (5) are not included. Therefore the complement is (\(-\infty,-6]\cup[5,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,-6]\cup[5,\infty\)). \(A\cup B=(-6,5)\), because (-6) and (5) are not included. Therefore the complement is (\(-\infty,-6]\cup[5,\infty\)).
Step 3
Exam Tip
\(A\cup B=(-6,5)\) है, क्योंकि (-6) और (5) शामिल नहीं हैं। इसलिए पूरक (\(-\infty,-6]\cup[5,\infty\)) है।
Taking complements reverses inclusion, so \(A'\subseteq B'\Rightarrow B\subseteq A\). Always remember this reversed order.
Step 2
Why this answer is correct
The correct answer is A. \(B\subseteq A\). Taking complements reverses inclusion, so \(A'\subseteq B'\Rightarrow B\subseteq A\). Always remember this reversed order.
Step 3
Exam Tip
पूरक लेने पर समावेशन उलट जाता है, इसलिए \(A'\subseteq B'\Rightarrow B\subseteq A\)। इस उल्टे क्रम को हमेशा याद रखें।
There are (25) odd numbers from (1) to (49), and the odd perfect squares are (1,9,25,49), so there are (4). Hence \(A'\cap B\) has (25-4=21) elements.
Step 2
Why this answer is correct
The correct answer is B. (21). There are (25) odd numbers from (1) to (49), and the odd perfect squares are (1,9,25,49), so there are (4). Hence \(A'\cap B\) has (25-4=21) elements.
Step 3
Exam Tip
(1) से (49) तक (25) विषम संख्याएँ हैं और विषम पूर्ण वर्ग (1,9,25,49) यानी (4) हैं। इसलिए \(A'\cap B\) में (25-4=21) अवयव हैं।
\(|x+2|\le 6\Rightarrow -8\le x\le 4\). Its complement is (x<-8) or (x>4), that is (\(-\infty,-8\)\cup\(4,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,-8\)\cup\(4,\infty\)). \(|x+2|\le 6\Rightarrow -8\le x\le 4\). Its complement is (x<-8) or (x>4), that is (\(-\infty,-8\)\cup\(4,\infty\)).
Step 3
Exam Tip
\(|x+2|\le 6\Rightarrow -8\le x\le 4\)। इसका पूरक (x<-8) या (x>4), यानी (\(-\infty,-8\)\cup\(4,\infty\)) है।
By inclusion-exclusion, \(|A\cup B\cup C|=30+20+12-10-6-4+2=44\). Hence the complement is (120-44=76).
Step 2
Why this answer is correct
The correct answer is B. (74). By inclusion-exclusion, \(|A\cup B\cup C|=30+20+12-10-6-4+2=44\). Hence the complement is (120-44=76).
Step 3
Exam Tip
समावेशन-बहिष्करण से \(|A\cup B\cup C|=30+20+12-10-6-4+2=44\)। इसलिए पूरक में (120-44=76) नहीं, ध्यान से सही गणना (30+20+12-10-6-4+2=44) और (120-44=76) है।
\(B-A=B\cap A'\), and if it equals (B), no element of (B) lies in (A). Hence \(A\cap B=\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(A\cap B=\varnothing\). \(B-A=B\cap A'\), and if it equals (B), no element of (B) lies in (A). Hence \(A\cap B=\varnothing\).
Step 3
Exam Tip
\(B-A=B\cap A'\) है और यह (B) के बराबर है, इसलिए (B) का कोई अवयव (A) में नहीं है। अतः \(A\cap B=\varnothing\)।
From \(A\subseteq B\), taking complements gives \(B'\subseteq A'\). The second condition is not needed for this conclusion.
Step 2
Why this answer is correct
The correct answer is A. \(B'\subseteq A'\). From \(A\subseteq B\), taking complements gives \(B'\subseteq A'\). The second condition is not needed for this conclusion.
Step 3
Exam Tip
\(A\subseteq B\) से पूरक लेने पर \(B'\subseteq A'\) मिलता है। दूसरी शर्त इस निष्कर्ष के लिए आवश्यक नहीं है।
(-1) is not in (A), (3) is in (A), and (8) is not in (A). Hence the complement is ([-1,3)\cup[8,\infty)).
Step 2
Why this answer is correct
The correct answer is A. ([-1,3)\cup[8,\infty)). (-1) is not in (A), (3) is in (A), and (8) is not in (A). Hence the complement is ([-1,3)\cup[8,\infty)).
Step 3
Exam Tip
(-1) (A) में नहीं है, (3) (A) में है और (8) (A) में नहीं है। इसलिए पूरक ([-1,3)\cup[8,\infty)) है।
\(A\cap A'\) is always \(\varnothing\), so it cannot be (U) when \(U\neq\varnothing\). A set and its complement are disjoint.
Step 2
Why this answer is correct
The correct answer is C. \(A\cap A'=U\). \(A\cap A'\) is always \(\varnothing\), so it cannot be (U) when \(U\neq\varnothing\). A set and its complement are disjoint.
Step 3
Exam Tip
\(A\cap A'\) हमेशा \(\varnothing\) होता है, इसलिए \(U\neq\varnothing\) होने पर यह (U) नहीं हो सकता। मूल समुच्चय और पूरक असंयुक्त होते हैं।
\(x^2-4x-12<0\Rightarrow -2<x<6\). Its complement is \(x\le -2\) or \(x\ge 6\), that is (\(-\infty,-2]\cup[6,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,-2]\cup[6,\infty\)). \(x^2-4x-12<0\Rightarrow -2<x<6\). Its complement is \(x\le -2\) or \(x\ge 6\), that is (\(-\infty,-2]\cup[6,\infty\)).
Step 3
Exam Tip
\(x^2-4x-12<0\Rightarrow -2<x<6\)। इसका पूरक \(x\le -2\) या \(x\ge 6\), यानी (\(-\infty,-2]\cup[6,\infty\)) है।
(B-A={9,15,18}), so its complement is (U-{9,15,18}). Option (A) gives exactly that set.
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,4,5,6,7,8,10,11,12,13,14,16,17}). (B-A={9,15,18}), so its complement is (U-{9,15,18}). Option (A) gives exactly that set.
Step 3
Exam Tip
(B-A={9,15,18}), इसलिए इसका पूरक (U-{9,15,18}) है। विकल्प (A) वही समुच्चय देता है।
A. अवयव दोनों में हैं या दोनों में नहीं हैं/Elements are in both sets or in neither set
Step 1
Concept
\(A\triangle B\) contains elements in exactly one set. Its complement gives elements that are in both or in neither.
Step 2
Why this answer is correct
The correct answer is A. अवयव दोनों में हैं या दोनों में नहीं हैं / Elements are in both sets or in neither set. \(A\triangle B\) contains elements in exactly one set. Its complement gives elements that are in both or in neither.
Step 3
Exam Tip
\(A\triangle B\) में वे अवयव होते हैं जो ठीक एक समुच्चय में हों। उसका पूरक वे अवयव देता है जो दोनों में हों या दोनों में न हों।
(A') is the set of odd numbers and (B') is the set of non-prime numbers. The odd non-primes from (1) to (30) are ({1,9,15,21,25,27}), so the count is (6).
Step 2
Why this answer is correct
The correct answer is A. (8). (A') is the set of odd numbers and (B') is the set of non-prime numbers. The odd non-primes from (1) to (30) are ({1,9,15,21,25,27}), so the count is (6).
Step 3
Exam Tip
(A') विषम संख्याएँ हैं और (B') अभाज्य नहीं संख्याएँ हैं। (1) से (30) तक विषम अभाज्य नहीं संख्याएँ ({1,9,15,21,25,27}) नहीं, साथ में (? ) नहीं; सही सूची ({1,9,15,21,25,27}) है, इसलिए संख्या (6) है।
By distributive law, (\(A\cup B'\)\cap\(A'\cup B'\)=B'\cup\(A\cap A'\)). Since \(A\cap A'=\varnothing\), the result is (B').
Step 2
Why this answer is correct
The correct answer is A. (B'\cup\(A\cap A'\)=B'). By distributive law, (\(A\cup B'\)\cap\(A'\cup B'\)=B'\cup\(A\cap A'\)). Since \(A\cap A'=\varnothing\), the result is (B').
Step 3
Exam Tip
वितरण नियम से (\(A\cup B'\)\cap\(A'\cup B'\)=B'\cup\(A\cap A'\))। चूंकि \(A\cap A'=\varnothing\), परिणाम (B') है।
\(A\cap B\) contains multiples of (\operatorname{lcm}(12,21)=84), so only (84) appears. Hence the complement has (84-1=83) elements.
Step 2
Why this answer is correct
The correct answer is B. (83). \(A\cap B\) contains multiples of (\operatorname{lcm}(12,21)=84), so only (84) appears. Hence the complement has (84-1=83) elements.
Step 3
Exam Tip
\(A\cap B\) में (\operatorname{lcm}(12,21)=84) के गुणज हैं, इसलिए केवल (84) आता है। अतः पूरक में (84-1=83) अवयव हैं।
(A) contains non-prime numbers, so (A') contains exactly prime numbers. (1) is not prime, so it is not in (A').
Step 2
Why this answer is correct
The correct answer is A. अभाज्य संख्याएँ / Prime numbers. (A) contains non-prime numbers, so (A') contains exactly prime numbers. (1) is not prime, so it is not in (A').
Step 3
Exam Tip
(A) में अभाज्य नहीं संख्याएँ हैं, इसलिए (A') में ठीक अभाज्य संख्याएँ होंगी। (1) अभाज्य नहीं है, इसलिए (A') में नहीं आएगा।
(A'\cup B'=\(A\cap B\)'), and \(A\cap B=[1,4]\). Therefore the complement is (\(-\infty,1\)\cup\(4,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,1\)\cup\(4,\infty\)). (A'\cup B'=\(A\cap B\)'), and \(A\cap B=[1,4]\). Therefore the complement is (\(-\infty,1\)\cup\(4,\infty\)).
Step 3
Exam Tip
(A'\cup B'=\(A\cap B\)') और \(A\cap B=[1,4]\) है। इसलिए पूरक (\(-\infty,1\)\cup\(4,\infty\)) होगा।
A. (U) सार्वत्रिक समुच्चय है/(U) is the universal set
Step 1
Concept
The complement of the universal set is always the empty set. Thus \(U'=\varnothing\) is a basic complement identity.
Step 2
Why this answer is correct
The correct answer is A. (U) सार्वत्रिक समुच्चय है / (U) is the universal set. The complement of the universal set is always the empty set. Thus \(U'=\varnothing\) is a basic complement identity.
Step 3
Exam Tip
सार्वत्रिक समुच्चय का पूरक हमेशा रिक्त समुच्चय होता है। इसलिए \(U'=\varnothing\) पूरक की मूल पहचान है।
(A') has (32-8=24) elements and \(B\subseteq A\), so \(A'\cap B=\varnothing\). (B) has (2) elements, hence the total is (26).
Step 2
Why this answer is correct
The correct answer is C. (26). (A') has (32-8=24) elements and \(B\subseteq A\), so \(A'\cap B=\varnothing\). (B) has (2) elements, hence the total is (26).
Step 3
Exam Tip
(A') में (32-8=24) अवयव हैं और \(B\subseteq A\), इसलिए \(A'\cap B=\varnothing\)। (B) में (2) अवयव हैं, अतः कुल (26) है।
Since \(C\subseteq A\) and \(C\subseteq B\), \(A\cup B\cup C=A\cup B\). \(|A\cup B|=18+12-6=24\), so the complement is (36-24=12).
Step 2
Why this answer is correct
The correct answer is C. (12). Since \(C\subseteq A\) and \(C\subseteq B\), \(A\cup B\cup C=A\cup B\). \(|A\cup B|=18+12-6=24\), so the complement is (36-24=12).
Step 3
Exam Tip
क्योंकि \(C\subseteq A\) और \(C\subseteq B\), इसलिए \(A\cup B\cup C=A\cup B\)। \(|A\cup B|=18+12-6=24\), अतः पूरक (36-24=12) है।
\(A\cup B=[-3,2\)\cup(2,6]), so (2) is not included. The complement contains (\(-\infty,-3\)), ({2}), and (\(6,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,-3\)\cup[2,2]\cup\(6,\infty\)). \(A\cup B=[-3,2\)\cup(2,6]), so (2) is not included. The complement contains (\(-\infty,-3\)), ({2}), and (\(6,\infty\)).
Step 3
Exam Tip
\(A\cup B=[-3,2\)\cup(2,6]) है, इसलिए (2) शामिल नहीं है। पूरक में (\(-\infty,-3\)), ({2}) और (\(6,\infty\)) आएंगे।
(A') is the set of even numbers, and (B') contains elements outside (B). Their common elements are ({4,6,10,12,14,16}).
Step 2
Why this answer is correct
The correct answer is A. ({4,6,10,12,14,16}). (A') is the set of even numbers, and (B') contains elements outside (B). Their common elements are ({4,6,10,12,14,16}).
Step 3
Exam Tip
(A') सम संख्याएँ हैं और (B') में (B) के बाहर के अवयव हैं। दोनों में सामान्य अवयव ({4,6,10,12,14,16}) हैं।
(-5) and (7) are in (A), while (-1) and (2) are not in (A). Therefore the complement is (\(-\infty,-5\)\cup[-1,2]\cup\(7,\infty\)).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,-5\)\cup[-1,2]\cup\(7,\infty\)). (-5) and (7) are in (A), while (-1) and (2) are not in (A). Therefore the complement is (\(-\infty,-5\)\cup[-1,2]\cup\(7,\infty\)).
Step 3
Exam Tip
(-5) और (7) (A) में हैं, जबकि (-1) और (2) (A) में नहीं हैं। इसलिए पूरक (\(-\infty,-5\)\cup[-1,2]\cup\(7,\infty\)) है।
(A'\cap B'=\(A\cup B\)'). If this is (U), the complement of \(A\cup B\) is (U), so \(A\cup B=\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(A\cup B=\varnothing\). (A'\cap B'=\(A\cup B\)'). If this is (U), the complement of \(A\cup B\) is (U), so \(A\cup B=\varnothing\).
Step 3
Exam Tip
(A'\cap B'=\(A\cup B\)') है। यदि यह (U) है, तो \(A\cup B\) का पूरक (U) है, इसलिए \(A\cup B=\varnothing\)।
From (1) to (33), there are (16) even numbers, and (6,12,18,24,30) are divisible by (3). So even elements in (A') are (16-5=11).
Step 2
Why this answer is correct
The correct answer is B. (11). From (1) to (33), there are (16) even numbers, and (6,12,18,24,30) are divisible by (3). So even elements in (A') are (16-5=11).
Step 3
Exam Tip
(1) से (33) तक (16) सम संख्याएँ हैं और उनमें (6,12,18,24,30) (3) से विभाज्य हैं। इसलिए (A') में सम अवयव (16-5=11) हैं।
एक समूह में (150) विद्यार्थी हैं, (82) गणित पढ़ते हैं, (74) जीवविज्ञान पढ़ते हैं और (36) दोनों पढ़ते हैं। दोनों में से कोई भी विषय न पढ़ने वाले कितने हैं?
\(|M\cup B|=82+74-36=120\), so the number studying neither is (|\(M\cup B\)'|=150-120=30). Read neither as the complement of the union.
Step 2
Why this answer is correct
The correct answer is B. (30). \(|M\cup B|=82+74-36=120\), so the number studying neither is (|\(M\cup B\)'|=150-120=30). Read neither as the complement of the union.
Step 3
Exam Tip
\(|M\cup B|=82+74-36=120\), इसलिए neither की संख्या (|\(M\cup B\)'|=150-120=30) है। neither को संघ के पूरक के रूप में पढ़ें।
