Topic Quiz • 150 questions • 35 seconds per question.
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If \(A=\{1,2,3\}\), how many elements will (\mathcal{P}(A)) have?
B. आठ8
तीन तत्वों वाले समुच्चय के उपसमुच्चय \(2^3=8\) होते हैं। परीक्षा में पहले (n(A)) गिनें। / A set with three elements has \(2^3=8\) subsets. In exams count (n(A)) first.
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If \(A=\{1,2,3,4\}\), how many elements will (\mathcal{P}(A)) have?
C. (16)
यदि समुच्चय में (n) तत्व हों, तो घात समुच्चय में \(2^n\) तत्व होते हैं। यहां \(2^4=16\), इसलिए यह नियम याद रखें। / If a set has (n) elements, its power set has \(2^n\) elements. Here \(2^4=16\), so remember this rule.
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If the universal set is \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{2,4,6,8,10\}\), what is (A')?
A. ({1,3,5,7,9})
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। परीक्षा में पूरक निकालते समय पहले सार्वत्रिक समुच्चय देखें। / (A') contains the elements of (U) that are not in (A). In exams, always check the universal set before finding a complement.
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If \(A=\{a,b\}\), which element will not belong to (\mathcal{P}(A))?
D. (a)
(\mathcal{P}(A)) के तत्व हमेशा (A) के उपसमुच्चय होते हैं। (a) स्वयं तत्व है उपसमुच्चय नहीं। / Elements of (\mathcal{P}(A)) are subsets of (A). The symbol (a) is an element not a subset.
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If \(A=\{a,b,{c}\}\), which of the following is an element of (\mathcal{P}(A))?
B. ({a,{c}})
घात समुच्चय का हर तत्व मूल समुच्चय का उपसमुच्चय होता है। ({a,{c}}) के दोनों तत्व (A) में हैं, इसलिए यह सही है। / Every element of a power set is a subset of the original set. Both elements of ({a,{c}}) are in (A), so it is correct.
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Let the universal set be \(U=\{1,2,3,4,5\}\) and \(A=\{2,4\}\). What is (A')?
A. {(1,3,5)}
पूरक में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। (U) बदले तो पूरक भी बदल सकता है। / The complement contains elements of (U) that are not in (A). If (U) changes the complement may change.
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If (n(\mathcal{P}(A))=32), how many proper subsets does (A) have?
C. (31)
\(2^n=32\) से (n=5), इसलिए कुल उपसमुच्चय (32) हैं। उचित उपसमुच्चय में स्वयं (A) नहीं आता, इसलिए संख्या (31) होगी। / From \(2^n=32\), we get (n=5), so total subsets are (32). A proper subset excludes (A) itself, so the number is (31).
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If (n(A)=4), how many proper subsets does (A) have?
B. (15)
कुल उपसमुच्चय \(2^4=16\) हैं और उचित उपसमुच्चय में (A) स्वयं शामिल नहीं होता। इसलिए संख्या (15) है। / Total subsets are \(2^4=16\) and a proper subset excludes (A) itself. Therefore the number is (15).
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B. (17)
(n\(A\cup B\)=18+22-7=33) होगा। बाहर के विद्यार्थी (50-33=17) हैं। / (n\(A\cup B\)=18+22-7=33). Students outside are (50-33=17).
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If (\mathcal{P}(A)) has (32) elements, how many elements are in (A)?
B. (5)
यदि \(2^n=32\) है तो (n=5) होगा। घात समुच्चय की संख्या देखकर मूल समुच्चय का आकार निकालें। / If \(2^n=32\), then (n=5). Use the size of the power set to find the size of the original set.
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Which statement is correct about the power set of the empty set?
B. (\mathcal{P}\(\varnothing\)={\varnothing})
रिक्त समुच्चय का एक ही उपसमुच्चय होता है, वह स्वयं रिक्त समुच्चय है। इसलिए (\mathcal{P}\(\varnothing\)) में एक तत्व होता है। / The empty set has only one subset, the empty set itself. Therefore (\mathcal{P}\(\varnothing\)) has one element.
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If \(A={\emptyset}\), how many elements will (\mathcal{P}(A)) have?
B. (2)
(A) में एक तत्व है जो \(\emptyset\) है। इसलिए उपसमुच्चय \(2^1=2\) होंगे। / (A) has one element which is \(\emptyset\). Hence it has \(2^1=2\) subsets.
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If \(A\subseteq U\) and \(A'= \varnothing\), what is the correct conclusion?
B. (A=U)
(A') खाली होने का अर्थ है कि (U) में (A) के बाहर कोई तत्व नहीं है। इसलिए (A) और (U) समान हैं। / (A') being empty means there is no element of (U) outside (A). Hence (A) and (U) are equal.
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Which statement about a universal set is correct?
B. यह संदर्भ के सभी विचाराधीन तत्वों को रखता हैIt contains all elements under consideration
सार्वत्रिक समुच्चय संदर्भ के अनुसार सभी विचाराधीन तत्वों का समुच्चय होता है। इसे हमेशा प्रश्न के संदर्भ से पहचानें। / A universal set contains all elements under discussion in the given context. Always identify it from the question context.
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If \(U={x:x\) is a digit(}) and \(A=\{2,4,6,8\}\), which is (A')?
A. {(0,1,3,5,7,9)}
अंक (0) से (9) तक होते हैं। (A) के बाहर बचे अंक (0,1,3,5,7,9) हैं। / Digits are from (0) to (9). The digits outside (A) are (0,1,3,5,7,9).
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If \(A=\{p,q,r\}\), what is the relation of ({p,q}) and ({p,q,r}) with (\mathcal{P}(A))?
A. दोनों तत्व हैंBoth are elements
दोनों (A) के उपसमुच्चय हैं इसलिए दोनों (\mathcal{P}(A)) के तत्व हैं। घात समुच्चय में पूरा (A) भी शामिल होता है। / Both are subsets of (A), so both are elements of (\mathcal{P}(A)). The power set also contains (A) itself.
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If \(A=\{1,2\}\), which is a correct element of (\mathcal{P}(A))?
B. {(1)}
({1}) समुच्चय (A) का उपसमुच्चय है इसलिए यह (\mathcal{P}(A)) का तत्व है। एकल तत्व और एकल समुच्चय में अंतर रखें। / ({1}) is a subset of (A), so it is an element of (\mathcal{P}(A)). Keep a single element and a singleton set distinct.
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If \(A=\{k,l,m\}\), what is (n(\mathcal{P}(A)))?
C. (8)
समुच्चय (A) में तीन अलग तत्व हैं इसलिए (n(A)=3)। घात समुच्चय में \(2^3=8\) तत्व होंगे। / The set (A) has three distinct elements, so (n(A)=3). The power set has \(2^3=8\) elements.
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If \(U=\{1,2,3,4,5,6\}\) and \(A=\{1,3,5\}\), how many elements are in (A')?
B. (3)
(A') में (2,4,6) होंगे इसलिए तीन तत्व हैं। पूरक निकालते समय केवल (U) के अंदर देखें। / (A') contains (2,4,6), so it has three elements. While finding complement look only inside (U).
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If \(A=\{1,{2}\}\), how many subsets will (A) have?
C. (4)
(A) में दो तत्व हैं (1) और ({2})। इसलिए उपसमुच्चय \(2^2=4\) होंगे। / (A) has two elements (1) and ({2}). Therefore it has \(2^2=4\) subsets.
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If \(U=\{a,b,c,d\}\) and \(A=\{a,d\}\), how many elements will (\mathcal{P}(A')) have?
B. (4)
(A'={b,c}) है जिसमें दो तत्व हैं। इसलिए (\mathcal{P}(A')) में \(2^2=4\) तत्व होंगे। / (A'={b,c}) has two elements. Hence (\mathcal{P}(A')) has \(2^2=4\) elements.
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If \(A=\{2,4,6\}\), what is the relation of \(\emptyset\) with (\mathcal{P}(A))?
A. \(\emptyset\in\mathcal{P}(A)\)
\(\emptyset\) हर समुच्चय का उपसमुच्चय है इसलिए यह उसके घात समुच्चय का तत्व है। यह बहुत सामान्य परीक्षा भ्रम है। / \(\emptyset\) is a subset of every set, so it is an element of its power set. This is a common exam confusion.
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If \(U=\{1,2,3,4\}\) and \(A=\emptyset\), what is (A')?
B. (U)
रिक्त समुच्चय में (U) का कोई तत्व नहीं है इसलिए उसका पूरक पूरा (U) होता है। पूरक हमेशा संदर्भ पर निर्भर है। / The empty set contains no element of (U), so its complement is the whole (U). Complement always depends on the context.
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If (A) has (5) elements, how many singleton subsets are in (\mathcal{P}(A))?
A. (5)
एकल उपसमुच्चय प्रत्येक एक तत्व से बनता है इसलिए उनकी संख्या (n(A)) के बराबर होती है। यहां संख्या (5) है। / Each singleton subset is formed from one element, so their number equals (n(A)). Here the number is (5).
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If \(A=\{1,2,3,4\}\), how many two-element subsets are in (\mathcal{P}(A))?
B. (6)
दो तत्वों वाले उपसमुच्चय चुनने की संख्या \({}^4C_2=6\) है। निश्चित आकार के उपसमुच्चयों में संयोजन का प्रयोग करें। / The number of two-element subsets is \({}^4C_2=6\). Use combinations for subsets of fixed size.
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If \(A=\{1,2,3\}\), how many three-element subsets will be in (\mathcal{P}(A))?
B. (1)
तीन तत्वों वाला एक ही उपसमुच्चय (A) स्वयं है। पूरा समुच्चय भी घात समुच्चय का तत्व होता है। / There is only one three-element subset which is (A) itself. The whole set is also an element of the power set.
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If (\mathcal{P}(A)={\emptyset,{x}}), what is (A)?
B. {(x)}
घात समुच्चय में (A) स्वयं भी एक तत्व होता है। इसलिए यहां \(A=\{x\}\) है। / The power set contains (A) itself as an element. Hence here \(A=\{x\}\).
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If \(A=\{1,2\}\) and (B=\mathcal{P}(A)), what is (n(B))?
C. (4)
(A) में दो तत्व हैं इसलिए (\mathcal{P}(A)) में \(2^2=4\) तत्व हैं। (B) वही घात समुच्चय है। / (A) has two elements, so (\mathcal{P}(A)) has \(2^2=4\) elements. (B) is that power set.
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If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{2,4,6,8\}\), which element is in (A')?
C. (5)
(A') में (U) के वे तत्व होंगे जो (A) में नहीं हैं। उनमें (5) शामिल है। / (A') contains elements of (U) that are not in (A). The element (5) is included.
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Which set is equal to (\mathcal{P}({m,n}))?
B. {\(\emptyset,{m},{n},{m,n}\)}
दो तत्वों वाले समुच्चय का घात समुच्चय चार उपसमुच्चयों से बनता है। तत्वों को उपसमुच्चय की तरह लिखना जरूरी है। / The power set of a two-element set has four subsets. The elements must be written as subsets.
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If \(A=\{0,1,2\}\), which statement is correct for ({0,2})?
A. \({0,2}\in\mathcal{P}(A)\)
({0,2}) के सभी तत्व (A) में हैं इसलिए यह (A) का उपसमुच्चय है। अतः यह (\mathcal{P}(A)) का तत्व है। / All elements of ({0,2}) are in (A), so it is a subset of (A). Hence it is an element of (\mathcal{P}(A)).
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If \(A=\{1,2,3,4,5\}\), how many three-element subsets are in (\mathcal{P}(A))?
B. (10)
तीन तत्वों वाले उपसमुच्चयों की संख्या \({}^5C_3=10\) है। निश्चित आकार की गिनती में क्रम नहीं माना जाता। / The number of three-element subsets is \({}^5C_3=10\). Order is not considered in fixed-size subset counting.
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If \(U={x:x\in \mathbb{N}, x\leq 10}\) and \(A={x:x\) is prime(}), what is (A')?
A. {(1,4,6,8,9,10)}
(10) तक अभाज्य संख्याएं (2,3,5,7) हैं। (U) में इनके बाहर (1,4,6,8,9,10) बचते हैं। / The primes up to (10) are (2,3,5,7). Outside these in (U), the elements are (1,4,6,8,9,10).
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If \(A=\{a,b,c,d\}\), which is the four-element subset in (\mathcal{P}(A))?
D. {(a,b,c,d)}
चार तत्वों वाला एकमात्र उपसमुच्चय (A) स्वयं है। पूरा समुच्चय अपने घात समुच्चय का तत्व होता है। / The only four-element subset is (A) itself. A set is an element of its own power set.
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If \(U=\{0,1,2,3,4,5\}\), \(A=\{0,2,4\}\), and \(B=\{1,3,5\}\), what is (A') equal to?
B. (B)
(A) में सम तत्व हैं और (U) में बचे तत्व (1,3,5) हैं। ये (B) के बराबर हैं। / (A) has even elements and the remaining elements in (U) are (1,3,5). These are equal to (B).
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In which case will (\mathcal{P}(A)) have exactly (16) elements?
C. (n(A)=4)
\(2^4=16\) इसलिए (A) में चार तत्व होने चाहिए। घात समुच्चय का सूत्र \(2^n\) याद रखें। / \(2^4=16\), so (A) must have four elements. Remember the power set formula \(2^n\).
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If (A) has (6) elements, how many total elements will (\mathcal{P}(A)) have?
D. (64)
छह तत्वों वाले समुच्चय के उपसमुच्चय \(2^6=64\) होते हैं। हर तत्व के लिए शामिल या न शामिल दो विकल्प होते हैं। / A six-element set has \(2^6=64\) subsets. Each element has two choices included or not included.
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If \(U=\{1,2,3,4,5\}\), what is (U')?
C. \(\emptyset\)
सार्वत्रिक समुच्चय का पूरक उसी संदर्भ में रिक्त होता है। (U) के बाहर कोई विचाराधीन तत्व नहीं माना जाता। / The complement of the universal set is empty in that context. No considered element lies outside (U).
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Which option is a correct member of (\mathcal{P}\(\emptyset\))?
B. \(\emptyset\)
(\mathcal{P}\(\emptyset\)={\emptyset}) होता है इसलिए उसका सदस्य \(\emptyset\) है। \({\emptyset}\) पूरा घात समुच्चय है सदस्य नहीं। / (\mathcal{P}\(\emptyset\)={\emptyset}), so its member is \(\emptyset\). \({\emptyset}\) is the whole power set not its member.
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If \(U=\{a,b,c,e\}\) and \(A=\{a,b,c,d\}\), what problem occurs before finding (A')?
A. \(A\subseteq U\) नहीं है\(A\subseteq U\) is not true
पूरक सामान्यतः तब लिया जाता है जब \(A\subseteq U\) हो। यहां (d), (U) में नहीं है इसलिए समस्या है। / Complement is normally taken when \(A\subseteq U\). Here (d) is not in (U), so there is a problem.
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If \(A=\{1,2,3,4\}\), how many singleton subsets containing odd numbers are in (\mathcal{P}(A))?
B. (2)
विषम तत्व (1) और (3) हैं इसलिए एकल उपसमुच्चय ({1},{3}) होंगे। संख्या (2) है। / The odd elements are (1) and (3), so the singleton subsets are ({1},{3}). The number is (2).
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If \(A=\{x,y,z\}\), how many subsets in (\mathcal{P}(A)) contain (x)?
C. (4)
(x) को स्थिर रखें और (y,z) के लिए दो दो विकल्प होंगे। इसलिए संख्या \(2^2=4\) है। / Fix (x) and each of (y,z) has two choices. Therefore the number is \(2^2=4\).
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If \(U=\{1,2,3,4,5,6\}\) and \(A=\{1,2\}\), what is (n(A'))?
C. (4)
(U) में (6) और (A) में (2) तत्व हैं। बचे हुए (4) तत्व (A') में होंगे। / (U) has (6) elements and (A) has (2) elements. The remaining (4) elements will be in (A').
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If \(A=\{1,2,3\}\), how many non-empty subsets are in (\mathcal{P}(A))?
B. (7)
कुल उपसमुच्चय \(2^3=8\) हैं और खाली उपसमुच्चय एक है। इसलिए खाली नहीं उपसमुच्चय (7) हैं। / Total subsets are \(2^3=8\) and one subset is empty. Therefore non-empty subsets are (7).
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If \(A=\{a,b,c\}\), how many subsets in (\mathcal{P}(A)) do not contain (a)?
C. (4)
(a) को बाहर रखें और (b,c) के लिए शामिल या न शामिल दो विकल्प होंगे। संख्या \(2^2=4\) है। / Keep (a) out and (b,c) each have two choices included or not. The number is \(2^2=4\).
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If \(U=\{1,2,3,4,5,6,7\}\) and \(A=\{3,5,7\}\), what is the smallest number in (A')?
A. (1)
(A') में (1,2,4,6) होंगे। इनमें सबसे छोटी संख्या (1) है। / (A') contains (1,2,4,6). The smallest among these is (1).
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If \(A=\{1,2,3,4\}\), how many subsets in (\mathcal{P}(A)) have at least three elements?
B. (5)
तीन तत्वों वाले \({}^4C_3=4\) और चार तत्वों वाला (1) उपसमुच्चय है। कुल (5) हैं। / There are \({}^4C_3=4\) three-element subsets and (1) four-element subset. The total is (5).
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If (\mathcal{P}(A)) has (8) elements, how many non-empty subsets does (A) have?
C. (7)
\(2^n=8\) से (n=3) है और कुल उपसमुच्चय (8) हैं। खाली उपसमुच्चय हटाने पर (7) बचते हैं। / From \(2^n=8\), (n=3), and total subsets are (8). Removing the empty subset leaves (7).
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B. (4)
सप्ताह में सात दिन होते हैं और (A) में तीन दिन हैं। इसलिए पूरक में चार दिन होंगे। / A week has seven days and (A) has three days. Therefore the complement has four days.
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If \(A=\{1,2\}\) and \(U=\{1,2,3\}\), what is (\mathcal{P}(A'))?
A. {\(\emptyset,{3}\)}
पहले (A'={3}) मिलेगा। फिर (\mathcal{P}(A')={\emptyset,{3}}) होगा। / First (A'={3}). Then (\mathcal{P}(A')={\emptyset,{3}}).
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If \(A=\{2,4,6,8,10\}\), how many elements will (\mathcal{P}(A)) have?
C. (32)
(A) में (5) तत्व हैं, इसलिए घात समुच्चय में \(2^5=32\) तत्व होंगे। परीक्षा में पहले मूल समुच्चय के तत्व गिनें। / Set (A) has (5) elements, so its power set has \(2^5=32\) elements. In exams, first count the elements of the original set.
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If \(A=\{1,3,5\}\), how many subsets in (\mathcal{P}(A)) must contain (3)?
B. (4)
(3) को रखना निश्चित है और बाकी (1,5) चुने या छोड़े जा सकते हैं। इसलिए संख्या \(2^2=4\) होगी। / Keeping (3) is fixed and the remaining (1,5) may be chosen or not chosen. So the number is \(2^2=4\).
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If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{2,3,5,7\}\), what is (A')?
A. ({1,4,6,8})
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। इसलिए (1,4,6,8) सही पूरक हैं। / (A') contains the elements of (U) that are not in (A). Therefore (1,4,6,8) form the correct complement.
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If (n(\mathcal{P}(B))=128), how many proper subsets does (B) have?
C. (127)
कुल उपसमुच्चय (128) हैं और उचित उपसमुच्चय में पूरा समुच्चय नहीं गिना जाता। इसलिए संख्या (128-1=127) होगी। / There are (128) total subsets, and a proper subset does not include the whole set itself. So the number is (128-1=127).
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If \(A=\{a,{b},c\}\), which of the following is not an element of (\mathcal{P}(A))?
D. ({a,b})
(b) स्वयं (A) का तत्व नहीं है, बल्कि ({b}) तत्व है। इसलिए ({a,b}), (A) का उपसमुच्चय नहीं है। / (b) itself is not an element of (A); ({b}) is an element. So ({a,b}) is not a subset of (A).
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If \(A=\{p,q,r\}\), which of the following is an element of (\mathcal{P}(A))?
B. ({p,q})
घात समुच्चय का तत्व हमेशा मूल समुच्चय का उपसमुच्चय होता है। ({p,q}) में केवल (A) के तत्व हैं, इसलिए यह सही है। / An element of a power set is always a subset of the original set. ({p,q}) contains only elements of (A), so it is correct.
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If \(U={x:x\in \mathbb{N}, x\leq 15}\) and \(A={x:x\) is a multiple of (5)(}), what is (n(A'))?
D. (12)
\(A=\{5,10,15\}\) है, इसलिए (n(A)=3)। (U) में (15) तत्व हैं, अतः (n(A')=15-3=12)। / \(A=\{5,10,15\}\), so (n(A)=3). Since (U) has (15) elements, (n(A')=15-3=12).
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If \(A=\{1,2,{3}\}\), which of the following is not an element of (\mathcal{P}(A))?
D. ({1,3})
(3) स्वयं (A) का तत्व नहीं है, बल्कि ({3}) तत्व है। इसलिए ({1,3}) (A) का उपसमुच्चय नहीं है। / (3) itself is not an element of (A); ({3}) is an element. So ({1,3}) is not a subset of (A).
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How many elements are there in the power set of the empty set \(\varnothing\)?
B. (1)
रिक्त समुच्चय में (0) तत्व हैं, इसलिए (\mathcal{P}\(\varnothing\)) में \(2^0=1\) तत्व होता है। याद रखें कि \(\varnothing\) भी एक उपसमुच्चय है। / The empty set has (0) elements, so (\mathcal{P}\(\varnothing\)) has \(2^0=1\) element. Remember that \(\varnothing\) is also a subset.
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How many elements will (\mathcal{P}(\mathcal{P}\(\varnothing\))) have?
B. (2)
(\mathcal{P}\(\varnothing\)={\varnothing}) में (1) तत्व है। इसलिए उसके घात समुच्चय में \(2^1=2\) तत्व होंगे। / (\mathcal{P}\(\varnothing\)={\varnothing}) has (1) element. Therefore its power set has \(2^1=2\) elements.
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If the universal set is \(U=\{a,b,c,d,e\}\) and \(A=\{b,d\}\), what is (A')?
A. ({a,c,e})
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं होते। यहां वे तत्व (a,c,e) हैं। / (A') contains the elements of (U) that are not in (A). Here those elements are (a,c,e).
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If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{1,3,5,7\}\), how many elements are in (A')?
C. (4)
(A') में (2,4,6,8) हैं, इसलिए इसमें (4) तत्व हैं। पूरक हमेशा दिए गए सार्वत्रिक समुच्चय के सापेक्ष होता है। / (A') contains (2,4,6,8), so it has (4) elements. A complement is always taken with respect to the given universal set.
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B. (13)
(n\(A\cup B\)=18+15-6=27), इसलिए बाहर के विद्यार्थी (40-27=13) हैं। ऐसे प्रश्नों में सार्वत्रिक समुच्चय कुल विद्यार्थियों का होता है। / (n\(A\cup B\)=18+15-6=27), so students outside are (40-27=13). In such questions, the universal set is the total group.
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If (A) and (B) are subsets of (U), what is (\(A\cup B\)') equal to?
B. \(A'\cap B'\)
डी मॉर्गन नियम के अनुसार (\(A\cup B\)'=A'\cap B')। संघ के पूरक में केवल वे तत्व आते हैं जो दोनों से बाहर हों। / By De Morgan's law, (\(A\cup B\)'=A'\cap B'). The complement of a union contains elements outside both sets.
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If \(A\subseteq U\), what will ((A')') be equal to?
A. (A)
किसी समुच्चय के पूरक का पूरक वही समुच्चय होता है। इसे द्वि-पूरक नियम कहते हैं। / The complement of the complement of a set is the set itself. This is called the double complement law.
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If \(A\subseteq U\) and (A'=U), what is (A)?
B. \(\varnothing\)
यदि (A) का पूरक पूरा (U) है, तो (A) में कोई तत्व नहीं हो सकता। इसलिए \(A=\varnothing\) है। / If the complement of (A) is the whole (U), then (A) cannot contain any element. Hence \(A=\varnothing\).
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If (U) has (35) elements and (A) has (12) elements, how many elements will (A') have?
B. (23)
(n(A')=n(U)-n(A)=35-12=23)। पूरक की गणना में कुल से दिए हुए समुच्चय को घटाते हैं। / (n(A')=n(U)-n(A)=35-12=23). To find a complement, subtract the set size from the total size.
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If \(A={x:x\) is a positive divisor of (18)(}), what is (n(\mathcal{P}(A)))?
C. (64)
(18) के धनात्मक भाजक (1,2,3,6,9,18) हैं, इसलिए (n(A)=6)। अतः (n(\mathcal{P}(A))=26=64)। / The positive divisors of (18) are (1,2,3,6,9,18), so (n(A)=6). Hence (n(\mathcal{P}(A))=26=64).
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If \(A=\{5\}\), what is (\mathcal{P}(A))?
D. \({\varnothing,{5}}\)
एकल समुच्चय के उपसमुच्चय \(\varnothing\) और वही समुच्चय होते हैं। इसलिए (5) नहीं, बल्कि ({5}) घात समुच्चय का तत्व है। / The subsets of a singleton set are \(\varnothing\) and the set itself. So ({5}), not (5), is an element of the power set.
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If \({7}\in \mathcal{P}(A)\), which statement is definitely true?
A. \(7\in A\)
\({7}\in \mathcal{P}(A)\) का अर्थ है कि \({7}\subseteq A\)। इसलिए \(7\in A\) होना आवश्यक है। / \({7}\in \mathcal{P}(A)\) means \({7}\subseteq A\). Therefore \(7\in A\) must be true.
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If \(A={\varnothing,1}\), which of the following is an element of (\mathcal{P}(A))?
D. उपरोक्त सभीAll of these
\(\varnothing\), \({\varnothing}\), और ({1}) सभी (A) के उपसमुच्चय हैं। इसलिए ये सभी (\mathcal{P}(A)) के तत्व हैं। / \(\varnothing\), \({\varnothing}\), and ({1}) are all subsets of (A). Therefore all of them are elements of (\mathcal{P}(A)).
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For any set (A), which two elements are always present in (\mathcal{P}(A))?
B. \(\varnothing\) और (A)\(\varnothing\) and (A)
हर समुच्चय का रिक्त समुच्चय और स्वयं वह समुच्चय उपसमुच्चय होते हैं। इसलिए दोनों हमेशा उसके घात समुच्चय में होते हैं। / The empty set and the set itself are subsets of every set. Therefore both are always in its power set.
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If (A) has (3) elements, how many non-empty proper subsets of (A) are there?
B. (6)
कुल उपसमुच्चय \(2^3=8\) हैं। अरिक्त उचित उपसमुच्चयों के लिए \(\varnothing\) और (A) हटाने पर (6) बचते हैं। / Total subsets are \(2^3=8\). Removing \(\varnothing\) and (A) gives (6) non-empty proper subsets.
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If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) have exactly (2) elements?
C. (6)
ठीक (2) तत्व वाले उपसमुच्चयों की संख्या \(\binom{4}{2}=6\) होती है। घात समुच्चय के अंदर उपसमुच्चयों को उनके आकार से भी गिना जा सकता है। / The number of subsets with exactly (2) elements is \(\binom{4}{2}=6\). Inside a power set, subsets can also be counted by their size.
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If (A) has (5) elements, how many subsets of (A) will contain one fixed element?
C. (16)
एक निश्चित तत्व रखने के बाद बाकी (4) तत्व चुने या छोड़े जा सकते हैं। इसलिए संख्या \(2^4=16\) होगी। / After fixing one element, each of the remaining (4) elements may be chosen or not chosen. So the number is \(2^4=16\).
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If (A) has (6) elements, how many subsets of (A) do not contain a selected element?
B. (32)
चुना हुआ तत्व हटाने पर (5) तत्व बचते हैं। इनके उपसमुच्चय \(2^5=32\) होंगे। / After excluding the selected element, (5) elements remain. Their subsets are \(2^5=32\).
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If \(A\subseteq U\), (n(U)=60), (n(A)=28), and (n(A')=x), what is the value of (x)?
B. (32)
पूरक के तत्व (U) में होते हैं पर (A) में नहीं होते। इसलिए (x=60-28=32) है। / Elements of the complement are in (U) but not in (A). Therefore (x=60-28=32).
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If \(U={x:x\in \mathbb{N}, x\leq 12}\) and \(A={x:x\) is a multiple of (3)(}), what is (A')?
B. ({1,2,4,5,7,8,10,11})
(U) में (1) से (12) तक संख्याएं हैं और \(A=\{3,6,9,12\}\) है। इन्हें हटाने पर (A') मिलता है। / (U) contains numbers from (1) to (12), and \(A=\{3,6,9,12\}\). Removing these gives (A').
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If \(A=\varnothing\) and the universal set is (U), what is (A') equal to?
B. (U)
रिक्त समुच्चय में कोई तत्व नहीं होता, इसलिए (U) के सभी तत्व उसके पूरक में आते हैं। अतः \(\varnothing'=U\)। / The empty set has no elements, so all elements of (U) lie in its complement. Hence \(\varnothing'=U\).
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If \(A=\{1,2,3\}\), how many non-empty elements are there in (\mathcal{P}(A))?
C. (7)
कुल उपसमुच्चय \(2^3=8\) हैं और केवल \(\varnothing\) रिक्त है। इसलिए अरिक्त उपसमुच्चय (8-1=7) हैं। / Total subsets are \(2^3=8\), and only \(\varnothing\) is empty. So non-empty subsets are (8-1=7).
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If \(A=\{m,n,o,p\}\), how many singleton sets will be in (\mathcal{P}(A))?
B. (4)
एक-तत्वीय उपसमुच्चय हर मूल तत्व से एक बनता है। (4) तत्वों से (4) singleton subsets बनेंगे। / A singleton subset is formed from each original element. From (4) elements, there will be (4) singleton subsets.
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If \(A=\{1,2,3,4,5\}\), how many sets in (\mathcal{P}(A)) have exactly (3) elements?
B. (10)
ठीक (3) तत्व चुनने के तरीके \(\binom{5}{3}=10\) हैं। ऐसे हर चयन से (\mathcal{P}(A)) का एक तत्व बनता है। / The number of ways to choose exactly (3) elements is \(\binom{5}{3}=10\). Each such choice is an element of (\mathcal{P}(A)).
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If \(A=\{0,1\}\), which correctly represents (\mathcal{P}(A))?
A. \({\varnothing,{0},{1},{0,1}}\)
दो तत्वों वाले समुच्चय के (4) उपसमुच्चय होते हैं। घात समुच्चय में तत्व उपसमुच्चय होते हैं, अकेले (0) और (1) नहीं। / A two-element set has (4) subsets. In a power set, the elements are subsets, not the single numbers (0) and (1).
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If \(U=\{1,2,3,4,5,6\}\), \(A=\{1,2,3\}\), and \(B=\{3,4,5\}\), what is \(A'\cap B'\)?
A. ({6})
(A'={4,5,6}) और (B'={1,2,6}) हैं। इनका समान भाग ({6}) है। / (A'={4,5,6}) and (B'={1,2,6}). Their common part is ({6}).
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If \(U=\{a,b,c,d\}\), \(A=\{a,b\}\), and \(B=\{b,c\}\), what is (\(A\cap B\)')?
B. ({a,c,d})
\(A\cap B={b}\) है, इसलिए उसका पूरक (U) में (b) को छोड़कर बाकी तत्व हैं। अतः ({a,c,d}) सही है। / \(A\cap B={b}\), so its complement is all elements of (U) except (b). Hence ({a,c,d}) is correct.
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If (n(A)=4) and (n(B)=3), what is the ratio of (n(\mathcal{P}(A))) to (n(\mathcal{P}(B)))?
A. (2:1)
(n(\mathcal{P}(A))=16) और (n(\mathcal{P}(B))=8) हैं। अनुपात (16:8=2:1) होगा। / (n(\mathcal{P}(A))=16) and (n(\mathcal{P}(B))=8). The ratio is (16:8=2:1).
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A. (8)
(LEVEL) के अलग-अलग अक्षर (L,E,V) हैं, इसलिए (n(A)=3)। अतः घात समुच्चय में \(2^3=8\) तत्व होंगे। / The distinct letters of (LEVEL) are (L,E,V), so (n(A)=3). Therefore the power set has \(2^3=8\) elements.
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If \(A=\{1,1,2,2,3\}\) is considered as a set, what is (n(\mathcal{P}(A)))?
A. (8)
समुच्चय में दोहराए गए तत्व एक बार ही गिने जाते हैं, इसलिए \(A=\{1,2,3\}\)। इस कारण (n(\mathcal{P}(A))=23=8)। / Repeated elements are counted only once in a set, so \(A=\{1,2,3\}\). Hence (n(\mathcal{P}(A))=23=8).
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If \(A=\{a,b,c,d,e,f\}\), how many subsets with exactly (5) elements are in (\mathcal{P}(A))?
B. (6)
(6) तत्वों में से (5) चुनने के तरीके \(\binom{6}{5}=6\) हैं। हर चयन घात समुच्चय का एक तत्व है। / The number of ways to choose (5) elements from (6) is \(\binom{6}{5}=6\). Each choice is one element of the power set.
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If (A) has (256) total subsets, how many elements does (A) have?
C. (8)
कुल उपसमुच्चय \(2^n\) होते हैं और \(256=2^8\)। इसलिए (A) में (8) तत्व हैं। / Total subsets are \(2^n\), and \(256=2^8\). Therefore (A) has (8) elements.
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B. \(B'\subseteq A'\)
यदि \(A\subseteq B\) है, तो (B) के बाहर के तत्व (A) के बाहर भी होंगे। इसलिए \(B'\subseteq A'\) होता है। / If \(A\subseteq B\), then every element outside (B) is also outside (A). Hence \(B'\subseteq A'\).
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If \(A={x:x\in \mathbb{Z}, -2\leq x\leq 2}\), how many elements will (\mathcal{P}(A)) have?
C. (32)
\(A=\{-2,-1,0,1,2\}\), इसलिए (n(A)=5)। अतः (n(\mathcal{P}(A))=25=32)। / \(A=\{-2,-1,0,1,2\}\), so (n(A)=5). Therefore (n(\mathcal{P}(A))=25=32).
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If \(A=\{1,2,3,4\}\), how many elements of (\mathcal{P}(A)) contain (1) but not (4)?
B. (4)
(1) को रखना और (4) को हटाना निश्चित है। बचे (2,3) के लिए \(2^2=4\) विकल्प हैं। / Keeping (1) and excluding (4) are fixed. For the remaining (2,3), there are \(2^2=4\) choices.
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If \(A=\{a,b,c,d,e\}\), how many subsets of (A) contain at least (4) elements?
B. (6)
कम से कम (4) तत्व का अर्थ (4) या (5) तत्व है। संख्या \(\binom{5}{4}+\binom{5}{5}=5+1=6\) है। / At least (4) elements means (4) or (5) elements. The number is \(\binom{5}{4}+\binom{5}{5}=5+1=6\).
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If \(A=\{1,2,3,4,5,6\}\), how many subsets of (A) contain both (2) and (5)?
B. (16)
(2) और (5) को रखना निश्चित है, बाकी (4) तत्व स्वतंत्र हैं। इसलिए उपसमुच्चय \(2^4=16\) होंगे। / Keeping (2) and (5) is fixed, and the remaining (4) elements are free. Therefore the number of subsets is \(2^4=16\).
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If \(A=\{1,2,3,4,5\}\), how many subsets of (A) contain neither (1) nor (2)?
B. (8)
(1) और (2) को हटाने के बाद (3,4,5) बचते हैं। इनके \(2^3=8\) उपसमुच्चय होंगे। / After excluding (1) and (2), the elements (3,4,5) remain. Their subsets are \(2^3=8\).
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If \(A=\{x,y,z\}\), how many elements of (\mathcal{P}(A)) are also proper subsets of (A)?
C. (7)
(\mathcal{P}(A)) में कुल (8) उपसमुच्चय हैं। उचित उपसमुच्चय के लिए केवल (A) को हटाते हैं, इसलिए (7) बचते हैं। / (\mathcal{P}(A)) has (8) total subsets. For proper subsets, only (A) itself is removed, so (7) remain.
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If \(U=\{1,2,3,4,5,6,7,8,9\}\), \(A=\{1,4,7\}\), and \(B=\{2,4,8\}\), what is (\(A\cup B\)')?
A. ({3,5,6,9})
\(A\cup B={1,2,4,7,8}\) है। (U) से इन्हें हटाने पर ({3,5,6,9}) मिलता है। / \(A\cup B={1,2,4,7,8}\). Removing these from (U) gives ({3,5,6,9}).
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If \(A=\{1,2,3\}\) and \(B=\{3,4\}\), how many elements will (\mathcal{P}\(A\cup B\)) have?
B. (16)
\(A\cup B={1,2,3,4}\), इसलिए इसमें (4) तत्व हैं। अतः उसके घात समुच्चय में \(2^4=16\) तत्व होंगे। / \(A\cup B={1,2,3,4}\), so it has (4) elements. Therefore its power set has \(2^4=16\) elements.
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C. (14)
(20) के धनात्मक भाजक (1,2,4,5,10,20) हैं, इसलिए (n(A)=6)। (n(A')=20-6=14) होगा। / The positive divisors of (20) are (1,2,4,5,10,20), so (n(A)=6). Thus (n(A')=20-6=14).
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If \(A=\{0,2,4,6\}\), how many elements will (\mathcal{P}(A)) have?
C. (16)
(A) में (4) तत्व हैं इसलिए (\mathcal{P}(A)) में \(2^4=16\) तत्व होंगे। परीक्षा में पहले मूल समुच्चय के अलग-अलग तत्व गिनें। / Set (A) has (4) elements so (\mathcal{P}(A)) has \(2^4=16\) elements. In exams first count the distinct elements of the original set.
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If \(U=\{1,2,3,4,5,6,7,8\}\) and \(A=\{2,4,6,8\}\), what is (n(\mathcal{P}(A')))?
C. (16)
(A'={1,3,5,7}) है, इसलिए (n(A')=4)। अतः (n(\mathcal{P}(A'))=24=16) होगा। / (A'={1,3,5,7}), so (n(A')=4). Hence (n(\mathcal{P}(A'))=24=16).
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If (n(\mathcal{P}(A))=512), what is the value of (n(A))?
C. (9)
\(512=2^9\) इसलिए (n(A)=9) होगा। घात समुच्चय में हमेशा \(2^n\) नियम लगाएं। / \(512=2^9\) so (n(A)=9). Always use the \(2^n\) rule for power set questions.
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If a set (B) has (1024) total subsets, how many proper subsets does it have?
A. (1023)
उचित उपसमुच्चय में पूरा समुच्चय स्वयं शामिल नहीं होता। इसलिए संख्या (1024-1=1023) है। / A proper subset does not include the whole set itself. Therefore the number is (1024-1=1023).
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If \(A=\{r,s,t,u\}\), which of the following is an element of (\mathcal{P}(A))?
B. ({r,t})
घात समुच्चय का प्रत्येक तत्व मूल समुच्चय का उपसमुच्चय होता है। ({r,t}) में केवल (A) के तत्व हैं इसलिए यह सही है। / Every element of a power set is a subset of the original set. ({r,t}) contains only elements of (A) so it is correct.
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If \(A=\{0,{1},2\}\), which of the following is not an element of (\mathcal{P}(A))?
D. ({0,1})
(1) स्वयं (A) का तत्व नहीं है बल्कि ({1}) तत्व है। इसलिए ({0,1}), (A) का उपसमुच्चय नहीं है। / (1) itself is not an element of (A); ({1}) is an element. So ({0,1}) is not a subset of (A).
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If \(A={\varnothing,{\varnothing}}\), what is (n(\mathcal{P}(A)))?
C. (4)
(A) में दो अलग-अलग तत्व हैं इसलिए उसके घात समुच्चय में \(2^2=4\) तत्व होंगे। \(\varnothing\) और \({\varnothing}\) को अलग समझें। / Set (A) has two distinct elements so its power set has \(2^2=4\) elements. Treat \(\varnothing\) and \({\varnothing}\) as different.
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How many elements are there in (\mathcal{P}(\mathcal{P}({a})))?
B. (4)
(\mathcal{P}({a})) में (2) तत्व होते हैं। इसलिए उसके घात समुच्चय में \(2^2=4\) तत्व होंगे। / (\mathcal{P}({a})) has (2) elements. Therefore its power set has \(2^2=4\) elements.
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If \(U=\{a,b,c,d,e,f\}\) and \(A=\{a,c,f\}\), what is (A')?
A. ({b,d,e})
(A') में (U) के वे तत्व आते हैं जो (A) में नहीं हैं। यहां वे तत्व (b,d,e) हैं। / (A') contains elements of (U) that are not in (A). Here those elements are (b,d,e).
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If \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{1,4,9\}\), how many elements will (A') have?
C. (7)
(U) में (10) और (A) में (3) तत्व हैं। अतः (n(A')=10-3=7)। / (U) has (10) elements and (A) has (3) elements. Hence (n(A')=10-3=7).
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B. (25)
(n\(A\cup B\)=32+28-10=50) होगा। किसी में नहीं (75-50=25) विद्यार्थी हैं। / (n\(A\cup B\)=32+28-10=50). Students in neither activity are (75-50=25).
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If (A) and (B) are subsets of (U), what is (\(A\cap B\)') equal to?
B. \(A'\cup B'\)
डी मॉर्गन नियम के अनुसार (\(A\cap B\)'=A'\cup B')। प्रतिच्छेद के पूरक में कम से कम एक समुच्चय से बाहर वाले तत्व आते हैं। / By De Morgan's law, (\(A\cap B\)'=A'\cup B'). The complement of an intersection includes elements outside at least one set.
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If \(A\subseteq U\) and \(A'= \varnothing\), which conclusion is correct?
B. (A=U)
पूरक खाली है तो (U) में (A) के बाहर कोई तत्व नहीं है। इसलिए (A=U) होगा। / If the complement is empty, no element of (U) lies outside (A). Therefore (A=U).
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If (n(U)=80) and (n(A')=19), what is (n(A))?
C. (61)
(A) और (A') मिलकर पूरा (U) बनाते हैं। इसलिए (n(A)=80-19=61)। / (A) and (A') together form the whole (U). So (n(A)=80-19=61).
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If \(A={x:x\) is a positive divisor of (24)(}), what is (n(\mathcal{P}(A)))?
C. (256)
(24) के भाजक (1,2,3,4,6,8,12,24) हैं इसलिए (n(A)=8)। अतः (n(\mathcal{P}(A))=28=256)। / The divisors of (24) are (1,2,3,4,6,8,12,24), so (n(A)=8). Hence (n(\mathcal{P}(A))=28=256).
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If \(A=\{k\}\), which is (\mathcal{P}(A))?
C. \({\varnothing,{k}}\)
एकल समुच्चय के उपसमुच्चय \(\varnothing\) और वही समुच्चय होते हैं। इसलिए घात समुच्चय में ({k}) आएगा, केवल (k) नहीं। / The subsets of a singleton set are \(\varnothing\) and the set itself. So the power set contains ({k}), not just (k).
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If \({m,n}\in \mathcal{P}(A)\), which statement is definitely true?
A. \(m\in A\) और \(n\in A\)\(m\in A\) and \(n\in A\)
\({m,n}\in \mathcal{P}(A)\) का अर्थ है \({m,n}\subseteq A\)। इसलिए (m) और (n) दोनों (A) के तत्व होंगे। / \({m,n}\in \mathcal{P}(A)\) means \({m,n}\subseteq A\). Therefore both (m) and (n) are elements of (A).
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If \(A={\varnothing,{\varnothing}}\), which of the following is an element of (\mathcal{P}(A))?
D. उपरोक्त सभीAll of these
दिए गए तीनों विकल्प (A) के उपसमुच्चय हैं। इसलिए ये सभी (\mathcal{P}(A)) के तत्व हैं। / All three given options are subsets of (A). Therefore all of them are elements of (\mathcal{P}(A)).
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Which statement is always true for a set (A)?
C. \(\varnothing\in \mathcal{P}(A)\)
\(\varnothing\) हर समुच्चय का उपसमुच्चय है। इसलिए \(\varnothing\) हमेशा (\mathcal{P}(A)) का तत्व होगा। / \(\varnothing\) is a subset of every set. Therefore \(\varnothing\) is always an element of (\mathcal{P}(A)).
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If (A) has (4) elements, how many non-empty proper subsets does (A) have?
A. (14)
कुल उपसमुच्चय \(2^4=16\) हैं। \(\varnothing\) और (A) को हटाने पर (14) अरिक्त उचित उपसमुच्चय बचते हैं। / Total subsets are \(2^4=16\). Removing \(\varnothing\) and (A) leaves (14) non-empty proper subsets.
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If \(A=\{2,4,6,8,10\}\), how many sets in (\mathcal{P}(A)) have exactly (2) elements?
B. (10)
ठीक (2) तत्व चुनने की संख्या \(\binom{5}{2}=10\) है। हर चयन घात समुच्चय का एक तत्व बनता है। / The number of ways to choose exactly (2) elements is \(\binom{5}{2}=10\). Each choice becomes an element of the power set.
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If (A) has (7) elements, how many subsets of (A) must contain one fixed element?
B. (64)
एक तत्व को रखना तय है और बाकी (6) तत्व स्वतंत्र हैं। इसलिए उपसमुच्चयों की संख्या \(2^6=64\) होगी। / One element is fixed to be included and the remaining (6) elements are free. So the number of subsets is \(2^6=64\).
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If (A) has (5) elements, how many subsets of (A) do not contain two fixed elements?
B. (8)
दो निश्चित तत्व हटाने पर (3) तत्व बचते हैं। इनके उपसमुच्चय \(2^3=8\) होंगे। / After excluding two fixed elements, (3) elements remain. Their subsets are \(2^3=8\).
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C. (14)
\(A=\{4,8,12,16\}\) है इसलिए (n(A)=4)। (U) में (18) तत्व हैं अतः (n(A')=18-4=14)। / \(A=\{4,8,12,16\}\), so (n(A)=4). Since (U) has (18) elements, (n(A')=18-4=14).
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If \(A=\varnothing\), which statement about (\mathcal{P}(A)) is correct?
B. (\mathcal{P}(A)={\varnothing})
रिक्त समुच्चय का एकमात्र उपसमुच्चय वही रिक्त समुच्चय है। इसलिए (\mathcal{P}\(\varnothing\)={\varnothing})। / The only subset of the empty set is the empty set itself. Therefore (\mathcal{P}\(\varnothing\)={\varnothing}).
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If (A=U), which statement about (\mathcal{P}(A)) and (\mathcal{P}(U)) is correct?
A. (\mathcal{P}(A)=\mathcal{P}(U))
समान समुच्चयों के सभी उपसमुच्चय भी समान होते हैं। इसलिए उनके घात समुच्चय बराबर होंगे। / Equal sets have exactly the same subsets. Therefore their power sets are equal.
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If \(A=\varnothing\) and the universal set is \(U=\{2,4,6\}\), what is (A')?
B. ({2,4,6})
रिक्त समुच्चय में कोई तत्व नहीं होता। इसलिए उसका पूरक पूरा (U) होगा। / The empty set has no elements. Therefore its complement is the whole (U).
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If \(A=\{3,6,9,12\}\), how many non-empty elements are there in (\mathcal{P}(A))?
B. (15)
कुल उपसमुच्चय \(2^4=16\) हैं। केवल \(\varnothing\) रिक्त है इसलिए अरिक्त उपसमुच्चय (15) होंगे। / Total subsets are \(2^4=16\). Only \(\varnothing\) is empty so there are (15) non-empty subsets.
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If \(A=\{a,e,i,o,u\}\), how many singleton sets are in (\mathcal{P}(A))?
B. (5)
हर मूल तत्व से एक एक-तत्वीय उपसमुच्चय बनता है। (5) तत्वों से (5) singleton subsets बनेंगे। / Each original element forms one singleton subset. From (5) elements, there are (5) singleton subsets.
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If \(A=\{1,2,3,4,5,6\}\), how many sets in (\mathcal{P}(A)) have exactly (4) elements?
B. (15)
ठीक (4) तत्व चुनने की संख्या \(\binom{6}{4}=15\) है। ऐसे हर उपसमुच्चय को घात समुच्चय में गिना जाता है। / The number of ways to choose exactly (4) elements is \(\binom{6}{4}=15\). Every such subset is counted in the power set.
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If \(A=\{2,3\}\), which is the correct form of (\mathcal{P}(A))?
A. \({\varnothing,{2},{3},{2,3}}\)
दो तत्वों वाले समुच्चय के चार उपसमुच्चय होते हैं। घात समुच्चय में (2) और (3) अकेले नहीं बल्कि ({2}) और ({3}) आते हैं। / A two-element set has four subsets. In the power set, (2) and (3) appear as ({2}) and ({3}), not alone.
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If \(U=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6\}\), and \(B=\{4,6,8\}\), what is \(A'\cap B'\)?
A. ({1,3,5,7})
(A'={1,3,5,7,8}) और (B'={1,2,3,5,7}) हैं। इनका प्रतिच्छेद ({1,3,5,7}) है। / (A'={1,3,5,7,8}) and (B'={1,2,3,5,7}). Their intersection is ({1,3,5,7}).
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If \(U=\{p,q,r,s,t\}\), \(A=\{p,q,r\}\), and \(B=\{r,s\}\), what is (\(A\cap B\)')?
B. ({p,q,s,t})
\(A\cap B={r}\) है। (U) से (r) हटाने पर ({p,q,s,t}) मिलता है। / \(A\cap B={r}\). Removing (r) from (U) gives ({p,q,s,t}).
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If (n(A)=6) and (n(B)=4), what is (n(\mathcal{P}(A)) : n(\mathcal{P}(B)))?
B. (4:1)
(n(\mathcal{P}(A))=64) और (n(\mathcal{P}(B))=16) हैं। अनुपात (64:16=4:1) होगा। / (n(\mathcal{P}(A))=64) and (n(\mathcal{P}(B))=16). The ratio is (64:16=4:1).
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If \(A={x:x\in \mathbb{N}, 5<x<11}\), what is (n(\mathcal{P}(A)))?
B. (32)
\(A=\{6,7,8,9,10\}\) है इसलिए (n(A)=5)। अतः (n(\mathcal{P}(A))=25=32)। / \(A=\{6,7,8,9,10\}\), so (n(A)=5). Hence (n(\mathcal{P}(A))=25=32).
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B. (8)
(MOON) के अलग-अलग अक्षर (M,O,N) हैं इसलिए (n(A)=3)। अतः घात समुच्चय में \(2^3=8\) तत्व होंगे। / The distinct letters of (MOON) are (M,O,N), so (n(A)=3). Therefore the power set has \(2^3=8\) elements.
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If \(A=\{4,4,5,6,6,7\}\) is considered as a set, what is (n(\mathcal{P}(A)))?
B. (16)
समुच्चय में दोहराए गए तत्व एक बार गिने जाते हैं इसलिए \(A=\{4,5,6,7\}\)। इसलिए (n(\mathcal{P}(A))=24=16)। / Repeated elements are counted once in a set, so \(A=\{4,5,6,7\}\). Hence (n(\mathcal{P}(A))=24=16).
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If \(A=\{a,b,c,d,e,f,g\}\), how many subsets with exactly (6) elements are in (\mathcal{P}(A))?
B. (7)
(7) में से (6) तत्व चुनने के तरीके \(\binom{7}{6}=7\) हैं। हर चयन एक उपसमुच्चय है। / The number of ways to choose (6) elements from (7) is \(\binom{7}{6}=7\). Each choice is a subset.
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If a set (A) has (2048) total subsets, how many elements does (A) have?
C. (11)
कुल उपसमुच्चय \(2^n\) होते हैं और \(2048=2^{11}\)। इसलिए (A) में (11) तत्व होंगे। / Total subsets are \(2^n\), and \(2048=2^{11}\). Therefore (A) has (11) elements.
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If \(A\subseteq B\subseteq U\), which relation is correct for complements?
B. \(B'\subseteq A'\)
छोटे समुच्चय का पूरक बड़ा होता है। \(A\subseteq B\) होने पर \(B'\subseteq A'\) सही है। / The complement of the smaller set is larger. If \(A\subseteq B\), then \(B'\subseteq A'\) is correct.
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If \(U=\{2,4,6,8,10,12\}\) and (A'={4,10}), what is (A)?
A. ({2,6,8,12})
(A') के तत्व (A) में नहीं होते। (U) से (4,10) हटाने पर \(A=\{2,6,8,12\}\) मिलता है। / Elements of (A') are not in (A). Removing (4,10) from (U) gives \(A=\{2,6,8,12\}\).
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If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) contain (2) but not (5)?
B. (8)
(2) रखना और (5) हटाना निश्चित है। बचे (1,3,4) के लिए \(2^3=8\) विकल्प हैं। / Including (2) and excluding (5) are fixed. For the remaining (1,3,4), there are \(2^3=8\) choices.
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If \(A=\{p,q,r,s,t,u\}\), how many subsets of (A) contain at least (5) elements?
B. (7)
कम से कम (5) का अर्थ (5) या (6) तत्व है। संख्या \(\binom{6}{5}+\binom{6}{6}=6+1=7\) है। / At least (5) means (5) or (6) elements. The number is \(\binom{6}{5}+\binom{6}{6}=6+1=7\).
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If \(A=\{1,2,3,4,5,6,7\}\), how many subsets of (A) contain both (3) and (7)?
B. (32)
(3) और (7) को रखना तय है और बाकी (5) तत्व स्वतंत्र हैं। इसलिए संख्या \(2^5=32\) होगी। / Including (3) and (7) is fixed and the remaining (5) elements are free. So the number is \(2^5=32\).
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If \(A=\{1,2,3,4,5,6\}\), how many subsets of (A) contain none of (1), (2), and (3)?
B. (8)
(1,2,3) हटाने पर (4,5,6) बचते हैं। इनके \(2^3=8\) उपसमुच्चय होंगे। / After excluding (1,2,3), the elements (4,5,6) remain. Their subsets are \(2^3=8\).
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If \(A=\{g,h,i,j\}\), how many elements of (\mathcal{P}(A)) are also proper subsets of (A)?
B. (15)
(\mathcal{P}(A)) में कुल \(2^4=16\) तत्व हैं। उचित उपसमुच्चय के लिए केवल (A) को हटाते हैं, इसलिए (15) बचते हैं। / (\mathcal{P}(A)) has \(2^4=16\) elements. For proper subsets, remove only (A) itself, so (15) remain.
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If \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{1,3,5,7\}\), and \(B=\{2,3,6,7\}\), what is (\(A\cup B\)')?
A. ({4,8,9,10})
\(A\cup B={1,2,3,5,6,7}\) है। (U) से इसे हटाने पर ({4,8,9,10}) मिलता है। / \(A\cup B={1,2,3,5,6,7}\). Removing it from (U) gives ({4,8,9,10}).
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If \(A=\{2,4,6\}\) and \(B=\{6,8,10\}\), how many elements will (\mathcal{P}\(A\cup B\)) have?
B. (32)
\(A\cup B={2,4,6,8,10}\) है इसलिए इसमें (5) तत्व हैं। अतः घात समुच्चय में \(2^5=32\) तत्व होंगे। / \(A\cup B={2,4,6,8,10}\), so it has (5) elements. Hence its power set has \(2^5=32\) elements.
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C. (22)
(30) के भाजक (1,2,3,5,6,10,15,30) हैं इसलिए (n(A)=8)। अतः (n(A')=30-8=22)। / The divisors of (30) are (1,2,3,5,6,10,15,30), so (n(A)=8). Hence (n(A')=30-8=22).
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If \(A=\{1,2,3\}\), how many elements of (\mathcal{P}(A)) have at least one element?
B. (7)
कम से कम एक तत्व वाले उपसमुच्चय अरिक्त उपसमुच्चय होते हैं। कुल \(2^3=8\) में से \(\varnothing\) हटाने पर (7) बचते हैं। / Subsets with at least one element are non-empty subsets. From total \(2^3=8\), removing \(\varnothing\) leaves (7).
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