Since \(A \subseteq U\), all \(2^4=16\) members of (\mathcal{P}(A)) are subsets of (U). In exams, first check whether \(A \subseteq U\).
Step 2
Why this answer is correct
The correct answer is C. (16). Since \(A \subseteq U\), all \(2^4=16\) members of (\mathcal{P}(A)) are subsets of (U). In exams, first check whether \(A \subseteq U\).
Step 3
Exam Tip
क्योंकि \(A \subseteq U\), इसलिए (\mathcal{P}(A)) के सभी \(2^4=16\) सदस्य (U) के उपसमुच्चय हैं। परीक्षा में पहले \(A \subseteq U\) जांचें।
Here (A'=U-A={b,d}), so (|\mathcal{P}(A')|=22=4). In exams, always take complement with respect to the given universal set.
Step 2
Why this answer is correct
The correct answer is B. (4). Here (A'=U-A={b,d}), so (|\mathcal{P}(A')|=22=4). In exams, always take complement with respect to the given universal set.
Step 3
Exam Tip
यहां (A'=U-A={b,d}), इसलिए (|\mathcal{P}(A')|=22=4)। परीक्षा में पूरक हमेशा दिए गए सार्वत्रिक समुच्चय के सापेक्ष लें।
After fixing (a), an even number of elements must be chosen from the remaining (4) elements for odd cardinality. The number of such choices is \(2^{4-1}=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). After fixing (a), an even number of elements must be chosen from the remaining (4) elements for odd cardinality. The number of such choices is \(2^{4-1}=8\).
Step 3
Exam Tip
(a) fixed होने पर odd cardinality के लिए बाकी (4) तत्वों में even संख्या चुननी होगी। ऐसे choices \(2^{4-1}=8\) हैं।
(\mathcal{P}\(\varnothing\)={\varnothing}) has (1) member, so (\mathcal{P}(\mathcal{P}(A))) has \(2^1=2\) members. Treat the empty set and its power set separately.
Step 2
Why this answer is correct
The correct answer is B. (2). (\mathcal{P}\(\varnothing\)={\varnothing}) has (1) member, so (\mathcal{P}(\mathcal{P}(A))) has \(2^1=2\) members. Treat the empty set and its power set separately.
Step 3
Exam Tip
(\mathcal{P}\(\varnothing\)={\varnothing}) में (1) सदस्य है, इसलिए (\mathcal{P}(\mathcal{P}(A))) में \(2^1=2\) सदस्य होंगे। खाली समुच्चय और उसके घात समुच्चय को अलग समझें।
(|\mathcal{P}(A)|=64), so (\mathcal{P}(\mathcal{P}(A))) has \(2^{64}\) members. In exams, do not reduce the exponent when taking a power set again.
Step 2
Why this answer is correct
The correct answer is C. \(2^{64}\). (|\mathcal{P}(A)|=64), so (\mathcal{P}(\mathcal{P}(A))) has \(2^{64}\) members. In exams, do not reduce the exponent when taking a power set again.
Step 3
Exam Tip
(|\mathcal{P}(A)|=64), इसलिए (\mathcal{P}(\mathcal{P}(A))) में \(2^{64}\) सदस्य होंगे। परीक्षा में घात समुच्चय पर फिर घात समुच्चय लगाने में घातांक न घटाएं।
({2}) itself is an element of (A), so ({{2}}) is a subset of (A). In exams, distinguish an element from a singleton set.
Step 2
Why this answer is correct
The correct answer is B. ({{2}}). ({2}) itself is an element of (A), so ({{2}}) is a subset of (A). In exams, distinguish an element from a singleton set.
Step 3
Exam Tip
({2}) स्वयं (A) का एक तत्व है, इसलिए ({{2}}) (A) का उपसमुच्चय है। परीक्षा में तत्व और एकल समुच्चय में फर्क करें।
The set (A) has two distinct members \(\varnothing\) and \({\varnothing}\), so (|\mathcal{P}(A)|=22=4). In exams, be careful while counting nested elements.
Step 2
Why this answer is correct
The correct answer is C. (4). The set (A) has two distinct members \(\varnothing\) and \({\varnothing}\), so (|\mathcal{P}(A)|=22=4). In exams, be careful while counting nested elements.
Step 3
Exam Tip
समुच्चय (A) में दो अलग सदस्य \(\varnothing\) और \({\varnothing}\) हैं, इसलिए (|\mathcal{P}(A)|=22=4)। परीक्षा में nested elements को गिनते समय सावधान रहें।
Since \(A\in \mathcal{P}(A)\), \(A\in \mathcal{P}(U)\), hence \(A\subseteq U\). In exams, infer \(A\subseteq U\) from \(\mathcal{P}(A)\subseteq \mathcal{P}(U)\).
Step 2
Why this answer is correct
The correct answer is A. \(A\subseteq U\). Since \(A\in \mathcal{P}(A)\), \(A\in \mathcal{P}(U)\), hence \(A\subseteq U\). In exams, infer \(A\subseteq U\) from \(\mathcal{P}(A)\subseteq \mathcal{P}(U)\).
Step 3
Exam Tip
क्योंकि \(A\in \mathcal{P}(A)\), इसलिए \(A\in \mathcal{P}(U)\) और अतः \(A\subseteq U\)। परीक्षा में \(\mathcal{P}(A)\subseteq \mathcal{P}(U)\) से \(A\subseteq U\) निकालें।
\(A\cup B={1,2,3}\), so total subsets are \(2^3=8\) and proper subsets are (8-1=7). In exams, do not forget to exclude the whole set.
Step 2
Why this answer is correct
The correct answer is B. (7). \(A\cup B={1,2,3}\), so total subsets are \(2^3=8\) and proper subsets are (8-1=7). In exams, do not forget to exclude the whole set.
Step 3
Exam Tip
\(A\cup B={1,2,3}\), इसलिए कुल subsets \(2^3=8\) और proper subsets (8-1=7) हैं। परीक्षा में पूरा समुच्चय हटाना न भूलें।
(A) and (A') are disjoint, so the only common subset is \(\varnothing\). In exams, understand intersection of power sets as common subsets.
Step 2
Why this answer is correct
The correct answer is B. \({\varnothing}\). (A) and (A') are disjoint, so the only common subset is \(\varnothing\). In exams, understand intersection of power sets as common subsets.
Step 3
Exam Tip
(A) और (A') असंबद्ध हैं, इसलिए दोनों के समान उपसमुच्चय में केवल \(\varnothing\) है। परीक्षा में intersection of power sets को common subsets समझें।
(\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}\(A\cap B\)), so the number is \(2^2=4\). This identity is very useful in exams.
Step 2
Why this answer is correct
The correct answer is B. (4). (\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}\(A\cap B\)), so the number is \(2^2=4\). This identity is very useful in exams.
Step 3
Exam Tip
(\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}\(A\cap B\)), इसलिए संख्या \(2^2=4\) है। परीक्षा में यह identity बहुत उपयोगी है।
(|\mathcal{P}\(A\cup B\)|=16), \(|\mathcal{P}(A)\cup\mathcal{P}(B)|=4+4-1=7\), so the difference is (9). In exams, count \(\varnothing\) as common to both.
Step 2
Why this answer is correct
The correct answer is C. (9). (|\mathcal{P}\(A\cup B\)|=16), \(|\mathcal{P}(A)\cup\mathcal{P}(B)|=4+4-1=7\), so the difference is (9). In exams, count \(\varnothing\) as common to both.
Step 3
Exam Tip
(|\mathcal{P}\(A\cup B\)|=16), \(|\mathcal{P}(A)\cup\mathcal{P}(B)|=4+4-1=7\), इसलिए अंतर (9) है। परीक्षा में \(\varnothing\) दोनों में common गिनें।
For an (n)-element set, the number of odd subsets is \(2^{n-1}\), so \(2^6=64\). In exams, even and odd subsets are equal in number.
Step 2
Why this answer is correct
The correct answer is B. (64). For an (n)-element set, the number of odd subsets is \(2^{n-1}\), so \(2^6=64\). In exams, even and odd subsets are equal in number.
Step 3
Exam Tip
किसी (n)-तत्वीय समुच्चय में odd subsets की संख्या \(2^{n-1}\) होती है, अतः \(2^6=64\)। परीक्षा में even और odd subsets बराबर होते हैं।
The number of even cardinality subsets is \(2^{8-1}=128\). In exams, remember that for \(n\geq1\), even and odd subsets are equal.
Step 2
Why this answer is correct
The correct answer is B. (128). The number of even cardinality subsets is \(2^{8-1}=128\). In exams, remember that for \(n\geq1\), even and odd subsets are equal.
Step 3
Exam Tip
Even cardinality subsets की संख्या \(2^{8-1}=128\) है। परीक्षा में \(n\geq1\) होने पर even और odd subsets बराबर याद रखें।
(1) is fixed and (5) is excluded, so the remaining ({2,3,4}) gives \(2^3=8\) choices. In exams, separate fixed and forbidden elements.
Step 2
Why this answer is correct
The correct answer is B. (8). (1) is fixed and (5) is excluded, so the remaining ({2,3,4}) gives \(2^3=8\) choices. In exams, separate fixed and forbidden elements.
Step 3
Exam Tip
(1) fixed है और (5) excluded है, इसलिए शेष ({2,3,4}) से \(2^3=8\) choices मिलती हैं। परीक्षा में fixed और forbidden elements अलग करें।
(a) and (b) are fixed, and the remaining (4) elements give \(2^4=16\) choices. In exams, place compulsory elements first.
Step 2
Why this answer is correct
The correct answer is B. (16). (a) and (b) are fixed, and the remaining (4) elements give \(2^4=16\) choices. In exams, place compulsory elements first.
Step 3
Exam Tip
(a) और (b) fixed हैं, बाकी (4) तत्वों पर \(2^4=16\) विकल्प हैं। परीक्षा में compulsory elements को पहले रख दें।
Total subsets are \(2^6=64\), and subsets containing neither (2) nor (4) are \(2^4=16\), so the answer is (48). In exams, use complement method for at least one.
Step 2
Why this answer is correct
The correct answer is C. (48). Total subsets are \(2^6=64\), and subsets containing neither (2) nor (4) are \(2^4=16\), so the answer is (48). In exams, use complement method for at least one.
Step 3
Exam Tip
कुल subsets \(2^6=64\) हैं और (2,4) दोनों न होने वाले subsets \(2^4=16\) हैं, इसलिए उत्तर (48) है। परीक्षा में at least one के लिए complement method तेज है।
A subset containing ({1,2}) has (3) and (4) as optional elements, so \(2^2=4\). In exams, treat the required subset as fixed.
Step 2
Why this answer is correct
The correct answer is B. (4). A subset containing ({1,2}) has (3) and (4) as optional elements, so \(2^2=4\). In exams, treat the required subset as fixed.
Step 3
Exam Tip
जिस subset में ({1,2}) शामिल हो, उसमें (3) और (4) वैकल्पिक हैं, इसलिए \(2^2=4\)। परीक्षा में required subset को fixed block मानें।
\(A\cup B={1,2,3,5,7,8}\), so the complement is ({4,6}) and the power set size is \(2^2=4\). In exams, take union first and then complement.
Step 2
Why this answer is correct
The correct answer is B. (4). \(A\cup B={1,2,3,5,7,8}\), so the complement is ({4,6}) and the power set size is \(2^2=4\). In exams, take union first and then complement.
Step 3
Exam Tip
\(A\cup B={1,2,3,5,7,8}\), इसलिए complement ({4,6}) है और power set size \(2^2=4\)। परीक्षा में पहले union फिर complement लें।
The primes in (U) are ({2,3,5,7,11}), so (|A'|=12-5=7) and the value is \(2^7=128\). In exams, do not count (1) as prime.
Step 2
Why this answer is correct
The correct answer is C. (256). The primes in (U) are ({2,3,5,7,11}), so (|A'|=12-5=7) and the value is \(2^7=128\). In exams, do not count (1) as prime.
Step 3
Exam Tip
(U) में primes ({2,3,5,7,11}) हैं, इसलिए (|A'|=12-5=7) और \(2^7=128\) नहीं? ध्यान दें (A') में (7) तत्व हैं, इसलिए सही मान (128) है।
(A') contains odd numbers ({1,3,5,7,9}), so non-empty subsets are \(2^5-1=31\). In exams, subtract (1) to exclude \(\varnothing\).
Step 2
Why this answer is correct
The correct answer is B. (31). (A') contains odd numbers ({1,3,5,7,9}), so non-empty subsets are \(2^5-1=31\). In exams, subtract (1) to exclude \(\varnothing\).
Step 3
Exam Tip
(A') में odd numbers ({1,3,5,7,9}) हैं, इसलिए non-empty subsets \(2^5-1=31\) हैं। परीक्षा में \(\varnothing\) हटाने पर (1) घटाएं।
B. असत्य क्योंकि \(1\notin \mathcal{P}(A)\)/False because \(1\notin \mathcal{P}(A)\)
Step 1
Concept
\({1}\subseteq \mathcal{P}(A)\) means \(1\in\mathcal{P}(A)\), which is false. In exams, read the meanings of \(\in\) and \(\subseteq\) carefully.
Step 2
Why this answer is correct
The correct answer is B. असत्य क्योंकि \(1\notin \mathcal{P}(A)\) / False because \(1\notin \mathcal{P}(A)\). \({1}\subseteq \mathcal{P}(A)\) means \(1\in\mathcal{P}(A)\), which is false. In exams, read the meanings of \(\in\) and \(\subseteq\) carefully.
Step 3
Exam Tip
\({1}\subseteq \mathcal{P}(A)\) का अर्थ है \(1\in\mathcal{P}(A)\), जो गलत है। परीक्षा में \(\in\) और \(\subseteq\) का अर्थ ध्यान से पढ़ें।
When (a) is excluded, the remaining (3) elements are free, so there are \(2^3=8\) subsets. In exams, remove the excluded element and count.
Step 2
Why this answer is correct
The correct answer is B. (8). When (a) is excluded, the remaining (3) elements are free, so there are \(2^3=8\) subsets. In exams, remove the excluded element and count.
Step 3
Exam Tip
(a) excluded होने पर शेष (3) तत्व स्वतंत्र हैं, इसलिए \(2^3=8\) subsets हैं। परीक्षा में excluded element हटाकर गिनें।
For the pair ({2,3}), there are (2) choices and for the remaining (3) elements there are \(2^3\), so \(2\cdot2^3=16\). In exams, split linked elements into cases.
Step 2
Why this answer is correct
The correct answer is B. (16). For the pair ({2,3}), there are (2) choices and for the remaining (3) elements there are \(2^3\), so \(2\cdot2^3=16\). In exams, split linked elements into cases.
Step 3
Exam Tip
जोड़ी ({2,3}) के लिए (2) choices हैं और बाकी (3) तत्वों के लिए \(2^3\), इसलिए \(2\cdot2^3=16\)। परीक्षा में linked elements को cases में बांटें।
After fixing (1), an odd number must be chosen from the remaining (5) elements, which is \(2^{5-1}=16\). In exams, adjust total parity using the fixed element.
Step 2
Why this answer is correct
The correct answer is B. (16). After fixing (1), an odd number must be chosen from the remaining (5) elements, which is \(2^{5-1}=16\). In exams, adjust total parity using the fixed element.
Step 3
Exam Tip
(1) fixed होने के बाद बाकी (5) तत्वों में odd number चुनना होगा, जिसकी संख्या \(2^{5-1}=16\) है। परीक्षा में total parity को fixed element से adjust करें।
\(A\cap B={3}\), so the complement has (5) elements and the power set size is \(2^5=32\). In exams, take complement from the universal set.
Step 2
Why this answer is correct
The correct answer is B. (32). \(A\cap B={3}\), so the complement has (5) elements and the power set size is \(2^5=32\). In exams, take complement from the universal set.
Step 3
Exam Tip
\(A\cap B={3}\), इसलिए complement में (5) तत्व हैं और power set size \(2^5=32\) है। परीक्षा में complement सार्वत्रिक समुच्चय से लें।
\(|A\cup B|=6+5-3=8\), so the complement has (2) elements and the power set size is \(2^2=4\). In exams, use inclusion-exclusion.
Step 2
Why this answer is correct
The correct answer is B. (4). \(|A\cup B|=6+5-3=8\), so the complement has (2) elements and the power set size is \(2^2=4\). In exams, use inclusion-exclusion.
Step 3
Exam Tip
\(|A\cup B|=6+5-3=8\), इसलिए complement में (2) तत्व हैं और power set size \(2^2=4\) है। परीक्षा में inclusion-exclusion लगाएं।
(\mathcal{P}(A)) has \(2^3=8\) members, so its power set has (8) singleton members. In exams, the number of singleton members equals the cardinality of the base set.
Step 2
Why this answer is correct
The correct answer is B. (8). (\mathcal{P}(A)) has \(2^3=8\) members, so its power set has (8) singleton members. In exams, the number of singleton members equals the cardinality of the base set.
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^3=8\) सदस्य हैं, इसलिए उसके power set में singleton members भी (8) होंगे। परीक्षा में singleton members की संख्या base set की cardinality के बराबर होती है।
Disjoint subsets are formed only from ({3,4}), so the number is \(2^2=4\). In exams, remove forbidden elements for a disjoint condition.
Step 2
Why this answer is correct
The correct answer is B. (4). Disjoint subsets are formed only from ({3,4}), so the number is \(2^2=4\). In exams, remove forbidden elements for a disjoint condition.
Step 3
Exam Tip
Disjoint subsets केवल ({3,4}) से बनेंगे, इसलिए संख्या \(2^2=4\) है। परीक्षा में disjoint condition के लिए forbidden elements हटाएं।
Total subsets are \(2^5=32\), and those of sizes (0,1,2) are (1+5+10=16), so the answer is (16). In exams, use complement counting.
Step 2
Why this answer is correct
The correct answer is A. (16). Total subsets are \(2^5=32\), and those of sizes (0,1,2) are (1+5+10=16), so the answer is (16). In exams, use complement counting.
Step 3
Exam Tip
कुल \(2^5=32\) subsets हैं और sizes (0,1,2) वाले (1+5+10=16) हैं, इसलिए उत्तर (16) है। परीक्षा में complement counting करें।
(1) is fixed, so the remaining (2) elements are chosen from (5): \(\binom{5}{2}=10\). In exams, count remaining choices after fixing the compulsory element.
Step 2
Why this answer is correct
The correct answer is B. (10). (1) is fixed, so the remaining (2) elements are chosen from (5): \(\binom{5}{2}=10\). In exams, count remaining choices after fixing the compulsory element.
Step 3
Exam Tip
(1) fixed है, इसलिए बाकी (2) तत्व (5) में से चुने जाएंगे: \(\binom{5}{2}=10\)। परीक्षा में compulsory element के बाद remaining choice गिनें।
After removing (1), (5) elements remain and (3) must be chosen: \(\binom{5}{3}=10\). In exams, remove the excluded element first.
Step 2
Why this answer is correct
The correct answer is A. (10). After removing (1), (5) elements remain and (3) must be chosen: \(\binom{5}{3}=10\). In exams, remove the excluded element first.
Step 3
Exam Tip
(1) हटाने पर (5) तत्व बचते हैं, जिनमें से (3) चुनने हैं: \(\binom{5}{3}=10\)। परीक्षा में excluded element को पहले हटा दें।
(A) is fixed and the (4) elements of (U-A) are optional, so \(2^4=16\). In exams, containing (A) means \(A\subseteq S\).
Step 2
Why this answer is correct
The correct answer is B. (16). (A) is fixed and the (4) elements of (U-A) are optional, so \(2^4=16\). In exams, containing (A) means \(A\subseteq S\).
Step 3
Exam Tip
(A) fixed है और (U-A) के (4) तत्व optional हैं, इसलिए \(2^4=16\)। परीक्षा में containing (A) का अर्थ \(A\subseteq S\) लें।
Subsets disjoint from (A') are exactly subsets of (A), so the number is \(2^4=16\). In exams, understand disjoint from complement as subset of the original set.
Step 2
Why this answer is correct
The correct answer is B. (16). Subsets disjoint from (A') are exactly subsets of (A), so the number is \(2^4=16\). In exams, understand disjoint from complement as subset of the original set.
Step 3
Exam Tip
(A') से disjoint subsets केवल (A) के subsets होंगे, इसलिए संख्या \(2^4=16\) है। परीक्षा में disjoint from complement का अर्थ subset of original set समझें।
