100 results found for "cardinality" in all classes.
यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनका cardinality (2) से अधिक है?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) have cardinality greater than (2)?
#sets
#cardinality
#power-set
A (16)
B (20)
C (26)
D (32)
Explanation opens after your attempt
Step 1
Concept
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।
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यदि (|A|=8) है, तो (\mathcal{P}(A)) में even cardinality वाले subsets कितने होंगे?
If (|A|=8), how many subsets in (\mathcal{P}(A)) have even cardinality?
#sets
#even-subsets
#cardinality
A (64)
B (128)
C (256)
D (512)
Explanation opens after your attempt
Step 1
Concept
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।
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(9) अवयवों वाले समुच्चय के विषम संख्या वाले उपसमुच्चयों की संख्या कितनी है?
How many subsets of odd cardinality does a set with (9) elements have?
#class11
#combinations
#subsets
#odd-cardinality
#expert
A (128)
B (256)
C (384)
D (512)
Explanation opens after your attempt
Step 1
Concept
Odd- and even-sized subsets are equal in number, so the count is \(2^8=256\). In such questions, half the subsets have odd size.
Step 2
Why this answer is correct
The correct answer is B. (256). Odd- and even-sized subsets are equal in number, so the count is \(2^8=256\). In such questions, half the subsets have odd size.
Step 3
Exam Tip
विषम और सम आकार के उपसमुच्चय बराबर होते हैं, इसलिए संख्या \(2^{8}=256\) है। ऐसे प्रश्नों में आधे उपसमुच्चय विषम आकार के होते हैं।
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यदि \(A=\{0,1\}\), \(B=\{2,3,4\}\) और \(C=\{4,5\}\) हैं, तो (\(A\times B\)\cup\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{0,1\}\), \(B=\{2,3,4\}\), and \(C=\{4,5\}\), what is the cardinality of (\(A\times B\)\cup\(A\times C\))?
#union-property
#set-identity
#cardinality
A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4,5}\). Thus there are \(2\cdot4=8\) elements.
Step 2
Why this answer is correct
The correct answer is B. (8). (\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4,5}\). Thus there are \(2\cdot4=8\) elements.
Step 3
Exam Tip
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)) और \(B\cup C={2,3,4,5}\)। इसलिए \(2\cdot4=8\) अवयव हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{1,3,5\}\) हैं, तो (\(A\times B\)\cap\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), and \(C=\{1,3,5\}\), what is the cardinality of (\(A\times B\)\cap\(A\times C\))?
#intersection-property
#cardinality
#set-identity
A (1)
B (3)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(3\cdot1=3\).
Step 2
Why this answer is correct
The correct answer is B. (3). (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(3\cdot1=3\).
Step 3
Exam Tip
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)) और \(B\cap C={3}\)। इसलिए कार्डिनलिटी \(3\cdot1=3\) है।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{5,6\}\) हैं, तो (\(A\times B\)\cup\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(C=\{5,6\}\), what is the cardinality of (\(A\times B\)\cup\(A\times C\))?
#union-property
#set-identity
#cardinality
A (6)
B (7)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={4,5,6}\). Thus there are \(3\cdot3=9\) elements.
Step 2
Why this answer is correct
The correct answer is C. (9). (\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={4,5,6}\). Thus there are \(3\cdot3=9\) elements.
Step 3
Exam Tip
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)) और \(B\cup C={4,5,6}\)। इसलिए \(3\cdot3=9\) अवयव हैं।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\) और \(C=\{3,4\}\) हैं, तो (\(A\times B\)\cap\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2\}\), \(B=\{2,3\}\), and \(C=\{3,4\}\), what is the cardinality of (\(A\times B\)\cap\(A\times C\))?
#intersection-property
#cardinality
#identity
A (0)
B (2)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(2\cdot1=2\).
Step 2
Why this answer is correct
The correct answer is B. (2). (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(2\cdot1=2\).
Step 3
Exam Tip
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)) और \(B\cap C={3}\)। इसलिए कार्डिनलिटी \(2\cdot1=2\) है।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\), \(C=\{3,4\}\) हैं, तो (\(A\times B\)\cup\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2\}\), \(B=\{2,3\}\), \(C=\{3,4\}\), what is the cardinality of (\(A\times B\)\cup\(A\times C\))?
#union-property
#cardinality
#set-identity
A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4}\). Hence the cardinality is \(2\cdot3=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). (\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4}\). Hence the cardinality is \(2\cdot3=6\).
Step 3
Exam Tip
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)) और \(B\cup C={2,3,4}\)। इसलिए कार्डिनलिटी \(2\cdot3=6\) है।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5,6\}\) हैं, तो (\(A\cup B\)\times C) की कार्डिनलिटी क्या है?
If \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5,6\}\), what is the cardinality of (\(A\cup B\)\times C)?
#union
#cardinality
#cartesian-product
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3,4}\), so (|\(A\cup B\)\times C|=4\cdot2=8). Find the union size first.
Step 2
Why this answer is correct
The correct answer is C. (8). \(A\cup B={1,2,3,4}\), so (|\(A\cup B\)\times C|=4\cdot2=8). Find the union size first.
Step 3
Exam Tip
\(A\cup B={1,2,3,4}\), इसलिए (|\(A\cup B\)\times C|=4\cdot2=8)। पहले संघ की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनकी cardinality (2) से विभाज्य है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) have cardinality divisible by (2)?
#sets
#even-subsets
#power-set
A (16)
B (32)
C (48)
D (64)
Explanation opens after your attempt
Step 1
Concept
Cardinality divisible by (2) means even cardinality. A (6)-element set has \(2^{6-1}=32\) even subsets.
Step 2
Why this answer is correct
The correct answer is B. (32). Cardinality divisible by (2) means even cardinality. A (6)-element set has \(2^{6-1}=32\) even subsets.
Step 3
Exam Tip
Cardinality (2) से विभाज्य होने का अर्थ even cardinality है। (6)-element set के even subsets \(2^{6-1}=32\) होते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में odd cardinality और (6) को रखने वाले subsets कितने हैं?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) have odd cardinality and contain (6)?
#sets
#parity
#counting
A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(6) is fixed, so an even number must be chosen from the remaining (5) elements. Such choices are \(2^{5-1}=16\).
Step 2
Why this answer is correct
The correct answer is B. (16). (6) is fixed, so an even number must be chosen from the remaining (5) elements. Such choices are \(2^{5-1}=16\).
Step 3
Exam Tip
(6) fixed है, इसलिए बाकी (5) तत्वों में even संख्या चुननी होगी। ऐसे choices \(2^{5-1}=16\) हैं।
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यदि (|A|=7) है, तो (\mathcal{P}(A)) में even cardinality वाले subsets की संख्या कितनी है?
If (|A|=7), how many subsets in (\mathcal{P}(A)) have even cardinality?
#sets
#even-subsets
#power-set
A (32)
B (64)
C (96)
D (128)
Explanation opens after your attempt
Step 1
Concept
When \(|A|=n\geq1\), the number of even cardinality subsets is \(2^{n-1}\). Here it is \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is B. (64). When \(|A|=n\geq1\), the number of even cardinality subsets is \(2^{n-1}\). Here it is \(2^6=64\).
Step 3
Exam Tip
जब \(|A|=n\geq1\), even cardinality subsets की संख्या \(2^{n-1}\) होती है। यहां \(2^6=64\) है।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनमें (1) हो और cardinality even हो?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) contain (1) and have even cardinality?
#sets
#parity
#power-set
A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
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।
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यदि \(A=\{1,2,3,4,5\}\), तो (\mathcal{P}(A)) के कितने तत्वों का आकार सम है?
If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) have even cardinality?
#sets
#power-set
#even-cardinality
A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
A set with (5) elements has (32) subsets. Half of them, (16), have even cardinality.
Step 2
Why this answer is correct
The correct answer is B. (16). A set with (5) elements has (32) subsets. Half of them, (16), have even cardinality.
Step 3
Exam Tip
(5) तत्वों वाले समुच्चय के कुल (32) उपसमुच्चय हैं। इनमें आधे यानी (16) सम आकार के होते हैं।
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कथन: यदि (n(A)=n(B)), तो (A=B)। कारण: समान सदस्य संख्या हमेशा समान सदस्य देती है। सही विकल्प चुनिए।
Assertion: If (n(A)=n(B)), then (A=B). Reason: Equal cardinality always gives identical elements. Choose the correct option.
#assertion-reason
#cardinality
#equal-sets
A कथन और कारण दोनों सही हैं / Both assertion and reason are true
B कथन सही है, कारण गलत है / Assertion is true, reason is false
C कथन गलत है, कारण सही है / Assertion is false, reason is true
D कथन और कारण दोनों गलत हैं / Both assertion and reason are false
Explanation opens after your attempt
Correct Answer
D. कथन और कारण दोनों गलत हैं / Both assertion and reason are false
Step 1
Concept
Equal cardinality does not force identical elements. For example, ({1,2}) and ({3,4}) are not equal.
Step 2
Why this answer is correct
The correct answer is D. कथन और कारण दोनों गलत हैं / Both assertion and reason are false. Equal cardinality does not force identical elements. For example, ({1,2}) and ({3,4}) are not equal.
Step 3
Exam Tip
सदस्य संख्या समान होने से सदस्य समान होना जरूरी नहीं है। जैसे ({1,2}) और ({3,4}) बराबर नहीं हैं।
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यदि (|A|=5) है, तो (\mathcal{P}(A)) में ऐसे members कितने हैं जिनमें किसी निश्चित तत्व \(a\in A\) को रखा गया है और cardinality odd है?
If (|A|=5), how many members of (\mathcal{P}(A)) contain a fixed element \(a\in A\) and have odd cardinality?
#sets
#power-set
#parity
#counting
A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
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।
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(10) तत्वों वाले समुच्चय के विषम संख्या तत्वों वाले उपसमुच्चयों की संख्या कितनी है?
How many subsets with an odd number of elements does a set of (10) elements have?
#subsets
#odd-cardinality
#identity
A (256)
B (500)
C (512)
D (1024)
Explanation opens after your attempt
Step 1
Concept
In an (n)-element set, the number of odd-sized subsets is \(2^{n-1}\). Here \(2^{9}=512\).
Step 2
Why this answer is correct
The correct answer is C. (512). In an (n)-element set, the number of odd-sized subsets is \(2^{n-1}\). Here \(2^{9}=512\).
Step 3
Exam Tip
किसी (n) तत्वों वाले समुच्चय में विषम आकार के उपसमुच्चय \(2^{n-1}\) होते हैं। यहां \(2^{9}=512\)।
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यदि \(A=\{1,2,3\}\) और \(B=\{a,b,c,d\}\) हों, तो (A) से (B) में फलन होने पर क्रमित युग्मों की संख्या कितनी होगी?
If \(A=\{1,2,3\}\) and \(B=\{a,b,c,d\}\), how many ordered pairs will a function from (A) to (B) have?
#relations-functions
#ordered-pairs
#domain-cardinality
A (3)
B (4)
C (7)
D (12)
Explanation opens after your attempt
Step 1
Concept
A function has exactly one pair for every domain element, so the number of pairs will be (3). In exams, pairs in a function equal the number of domain elements.
Step 2
Why this answer is correct
The correct answer is A. (3). A function has exactly one pair for every domain element, so the number of pairs will be (3). In exams, pairs in a function equal the number of domain elements.
Step 3
Exam Tip
फलन में प्रांत के हर तत्व के लिए ठीक एक युग्म होता है इसलिए युग्मों की संख्या (3) होगी। परीक्षा में फलन के युग्म प्रांत के तत्वों के बराबर होते हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5,6,7\}\) हों, तो (A) से (B) में फलन होने पर क्रमित युग्मों की संख्या कितनी होगी?
If \(A=\{1,2,3\}\) and \(B=\{4,5,6,7\}\), how many ordered pairs will a function from (A) to (B) have?
#relations-functions
#domain-cardinality
#ordered-pairs
A (3)
B (4)
C (7)
D (12)
Explanation opens after your attempt
Step 1
Concept
A function assigns exactly one image to every element of domain (A), so it will have (3) pairs. In exams, pairs in a function equal the number of domain elements.
Step 2
Why this answer is correct
The correct answer is A. (3). A function assigns exactly one image to every element of domain (A), so it will have (3) pairs. In exams, pairs in a function equal the number of domain elements.
Step 3
Exam Tip
फलन में प्रांत (A) के हर तत्व की ठीक एक छवि होती है, इसलिए युग्मों की संख्या (3) होगी। परीक्षा में फलन के युग्म प्रांत के तत्वों के बराबर होते हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{a,b,c\}\) हों, तो (A) से (B) में फलन होने के लिए किसी संबंध में कम से कम कितने क्रमित युग्म होने चाहिए?
If \(A=\{1,2,3,4\}\) and \(B=\{a,b,c\}\), how many ordered pairs must a relation have at minimum to be a function from (A) to (B)?
#relations-functions
#minimum-pairs
#domain-cardinality
A (3)
B (4)
C (7)
D (12)
Explanation opens after your attempt
Step 1
Concept
A function must assign exactly one image to each of the (4) elements of (A), so (4) pairs are needed. In exams, the number of pairs in a function equals the number of domain elements.
Step 2
Why this answer is correct
The correct answer is B. (4). A function must assign exactly one image to each of the (4) elements of (A), so (4) pairs are needed. In exams, the number of pairs in a function equals the number of domain elements.
Step 3
Exam Tip
फलन में (A) के हर (4) तत्व की ठीक एक छवि होनी चाहिए, इसलिए (4) युग्म चाहिए। परीक्षा में फलन के युग्मों की संख्या प्रांत के तत्वों के बराबर होती है।
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(\mathcal{P}(X)) पर (ARB) तब और केवल तब जब (|A|=|B|)। यह संबंध कैसा है?
On (\mathcal{P}(X)), (ARB) if and only if (|A|=|B|). What type of relation is it?
#relations
#power-set
#cardinality
A समतुल्यता संबंध / Equivalence relation
B केवल सममित / Only symmetric
C प्रतिवर्ती नहीं / Not reflexive
D संक्रामी नहीं / Not transitive
Explanation opens after your attempt
Correct Answer
A. समतुल्यता संबंध / Equivalence relation
Step 1
Concept
Every set has the same size as itself. Having equal size is also symmetric and transitive.
Step 2
Why this answer is correct
The correct answer is A. समतुल्यता संबंध / Equivalence relation. Every set has the same size as itself. Having equal size is also symmetric and transitive.
Step 3
Exam Tip
हर समुच्चय का आकार अपने बराबर होता है। बराबर आकार होना सममित और संक्रामी भी है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5\}\) और \(T=\{(a,b)\in A\times B:a^2-b\le3\}\), तो (n(T)) कितना है?
If \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5\}\), and \(T=\{(a,b)\in A\times B:a^2-b\le3\}\), what is (n(T))?
#cartesian-product
#inequality
#relation-cardinality
A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the valid numbers of (b) are (4,4,1,0). Total is (9), so apply the condition for each first coordinate.
Step 2
Why this answer is correct
The correct answer is B. (9). For (a=1,2,3,4), the valid numbers of (b) are (4,4,1,0). Total is (9), so apply the condition for each first coordinate.
Step 3
Exam Tip
(a=1,2,3,4) पर मान्य (b) की संख्याएं क्रमशः (4,4,1,0) हैं। कुल (9) युग्म हैं, इसलिए हर पहले घटक पर शर्त लगाएं।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\) और \(S=\{(a,b)\in A\times B:a+b\ge5\}\), तो (n(S)) कितना है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\), and \(S=\{(a,b)\in A\times B:a+b\ge5\}\), what is (n(S))?
#cartesian-product
#relation-cardinality
#inequality
A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the counts are (0,1,2,3), totaling (6). Do not forget equality in a boundary inequality.
Step 2
Why this answer is correct
The correct answer is C. (6). For (a=1,2,3,4), the counts are (0,1,2,3), totaling (6). Do not forget equality in a boundary inequality.
Step 3
Exam Tip
(a=1,2,3,4) पर मान क्रमशः (0,0,2,3) नहीं बल्कि (0,1,2,3) हैं, कुल (6) है। सीमा वाली असमता में बराबरी को न भूलें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5,6\}\), तो (\(A\cup B\)\times C) में कितने तत्व हैं?
If \(A=\{1,2\}\), \(B=\{3,4\}\), and \(C=\{5,6\}\), how many elements are in (\(A\cup B\)\times C)?
#cartesian-product
#union
#cardinality
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3,4}\) has (4) elements and (C) has (2) elements. Hence \(4\cdot2=8\) pairs are formed.
Step 2
Why this answer is correct
The correct answer is C. (8). \(A\cup B={1,2,3,4}\) has (4) elements and (C) has (2) elements. Hence \(4\cdot2=8\) pairs are formed.
Step 3
Exam Tip
\(A\cup B={1,2,3,4}\) में (4) तत्व हैं और (C) में (2) तत्व हैं। इसलिए \(4\cdot2=8\) युग्म बनेंगे।
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यदि (n(A)=4), (n(B)=5) और (n\(A\cap B\)=2), तो (n(\(A\cap B\)\times\(A\cup B\))) कितना है?
If (n(A)=4), (n(B)=5), and (n\(A\cap B\)=2), what is (n(\(A\cap B\)\times\(A\cup B\)))?
#cartesian-product
#union-intersection
#cardinality
A (10)
B (12)
C (14)
D (18)
Explanation opens after your attempt
Step 1
Concept
(n\(A\cup B\)=4+5-2=7) and (n\(A\cap B\)=2). Hence the total is \(2\cdot7=14\).
Step 2
Why this answer is correct
The correct answer is C. (14). (n\(A\cup B\)=4+5-2=7) and (n\(A\cap B\)=2). Hence the total is \(2\cdot7=14\).
Step 3
Exam Tip
(n\(A\cup B\)=4+5-2=7) और (n\(A\cap B\)=2)। इसलिए कुल \(2\cdot7=14\) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\), तो (P\(A\times B\)) में कितने तत्व होंगे?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), how many elements are in (P\(A\times B\))?
#cartesian-product
#power-set
#cardinality
A (6)
B (12)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=3\cdot2=6), so (n(P\(A\times B\))=26 =64). Use \(2^n\) for a power set.
Step 2
Why this answer is correct
The correct answer is D. (64). (n\(A\times B\)=3\cdot2=6), so (n(P\(A\times B\))=26 =64). Use \(2^n\) for a power set.
Step 3
Exam Tip
(n\(A\times B\)=3\cdot2=6), इसलिए (n(P\(A\times B\))=26 =64)। घात समुच्चय के लिए \(2^n\) लगाएं।
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यदि (A) में (m) तत्व और (B) में (n) तत्व हैं तथा (n\(A\times B\)=n\(B\times A\)), तो इस कथन के बारे में सही विकल्प चुनिए।
If (A) has (m) elements and (B) has (n) elements and (n\(A\times B\)=n\(B\times A\)), choose the correct statement.
#cartesian-product
#cardinality
#conceptual
A यह केवल (m=n) पर सत्य है / It is true only when (m=n)
B यह हमेशा सत्य है / It is always true
C यह केवल (A=B) पर सत्य है / It is true only when (A=B)
D यह कभी सत्य नहीं है / It is never true
Explanation opens after your attempt
Correct Answer
B. यह हमेशा सत्य है / It is always true
Step 1
Concept
(n\(A\times B\)=mn) and (n\(B\times A\)=nm), so they are equal. Equal cardinality does not necessarily mean equal sets.
Step 2
Why this answer is correct
The correct answer is B. यह हमेशा सत्य है / It is always true. (n\(A\times B\)=mn) and (n\(B\times A\)=nm), so they are equal. Equal cardinality does not necessarily mean equal sets.
Step 3
Exam Tip
(n\(A\times B\)=mn) और (n\(B\times A\)=nm), इसलिए दोनों बराबर हैं। बराबर संख्या का अर्थ बराबर समुच्चय होना जरूरी नहीं है।
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यदि (n\(A\times B\)=n\(B\times C\)=30), (n(B)=5), और (A,B,C) अशून्य सीमित समुच्चय हैं, तो (n\(A\times C\)) कितना होगा?
If (n\(A\times B\)=n\(B\times C\)=30), (n(B)=5), and (A,B,C) are non-empty finite sets, what is (n\(A\times C\))?
#relations-functions
#cartesian-product
#cardinality-chain
#expert
A (25)
B (30)
C (36)
D (60)
Explanation opens after your attempt
Step 1
Concept
(n(A)=30/5=6) and (n(C)=30/5=6). Therefore (n\(A\times C\)=6\cdot6=36).
Step 2
Why this answer is correct
The correct answer is C. (36). (n(A)=30/5=6) and (n(C)=30/5=6). Therefore (n\(A\times C\)=6\cdot6=36).
Step 3
Exam Tip
(n(A)=30/5=6) और (n(C)=30/5=6)। इसलिए (n\(A\times C\)=6\cdot6=36)।
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यदि \(A\times B\) में (24) अवयव हैं और (A) में (B) से (2) अधिक अवयव हैं, तो (n(A)) क्या है?
If \(A\times B\) has (24) elements and (A) has (2) more elements than (B), what is (n(A))?
#relations-functions
#cartesian-product
#cardinality-equation
#expert
A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
Let (n(B)=m), then (n(A)=m+2) and (m(m+2)=24). This gives (m=4), so (n(A)=6).
Step 2
Why this answer is correct
The correct answer is B. (6). Let (n(B)=m), then (n(A)=m+2) and (m(m+2)=24). This gives (m=4), so (n(A)=6).
Step 3
Exam Tip
मान लें (n(B)=m), तब (n(A)=m+2) और (m(m+2)=24)। इससे (m=4), इसलिए (n(A)=6)।
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यदि (n\(A\times B\)=72) और (n(A)=8) है, तो (n(B)) कितना होगा?
If (n\(A\times B\)=72) and (n(A)=8), then what is (n(B))?
#relations-functions
#cartesian-product
#cardinality
#expert
A (9)
B (8)
C (64)
D (80)
Explanation opens after your attempt
Step 1
Concept
Since (n\(A\times B\)=n(A)n(B)), (72=8n(B)). In exams, apply the product rule first.
Step 2
Why this answer is correct
The correct answer is A. (9). Since (n\(A\times B\)=n(A)n(B)), (72=8n(B)). In exams, apply the product rule first.
Step 3
Exam Tip
क्योंकि (n\(A\times B\)=n(A)n(B)), इसलिए (72=8n(B))। परीक्षा में पहले गुणन नियम लगाएं।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\) और \(C=\{2,3,5\}\) हैं, तो (|A\times(B-C)|+|\(A\times B\)\cap\(A\times C\)|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(C=\{2,3,5\}\), what is (|A\times(B-C)|+|\(A\times B\)\cap\(A\times C\)|)?
#set-operations
#cardinality
#cartesian-product
A (9)
B (12)
C (15)
D (18)
Explanation opens after your attempt
Step 1
Concept
(B-C={1,4}) gives (6) pairs, and \(B\cap C={2,3}\) gives (6) pairs. The sum is (12).
Step 2
Why this answer is correct
The correct answer is B. (12). (B-C={1,4}) gives (6) pairs, and \(B\cap C={2,3}\) gives (6) pairs. The sum is (12).
Step 3
Exam Tip
(B-C={1,4}) से (6) और \(B\cap C={2,3}\) से (6) युग्म मिलते हैं। योग (12) है।
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यदि \(|A\times B|=24\), (|A|=4) और (|C|=7) हैं, तो \(|B\times C|\) का मान क्या होगा?
If \(|A\times B|=24\), (|A|=4), and (|C|=7), what is the value of \(|B\times C|\)?
#cartesian-product
#cardinality
#expert
A (28)
B (35)
C (42)
D (56)
Explanation opens after your attempt
Step 1
Concept
Since \(|B|=\frac{24}{4}=6\), \(|B\times C|=6\cdot7=42\). First find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is C. (42). Since \(|B|=\frac{24}{4}=6\), \(|B\times C|=6\cdot7=42\). First find the unknown cardinality.
Step 3
Exam Tip
\(|B|=\frac{24}{4}=6\), इसलिए \(|B\times C|=6\cdot7=42\)। पहले अज्ञात समुच्चय की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2\}\), \(B=\{3,4,5\}\) और \(C=\{6,7,8\}\) हैं, तो (|\(A\times B\)\times C|) क्या है?
If \(A=\{1,2\}\), \(B=\{3,4,5\}\), and \(C=\{6,7,8\}\), what is (|\(A\times B\)\times C|)?
#nested-product
#cardinality
#ordered-pair
A (12)
B (15)
C (18)
D (24)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=2\cdot3=6\), and then \(6\cdot3=18\). Cardinality also multiplies in nested products.
Step 2
Why this answer is correct
The correct answer is C. (18). \(|A\times B|=2\cdot3=6\), and then \(6\cdot3=18\). Cardinality also multiplies in nested products.
Step 3
Exam Tip
\(|A\times B|=2\cdot3=6\) और फिर \(6\cdot3=18\)। नेस्टेड गुणनफल में भी कार्डिनलिटी गुणा होती है।
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यदि \(A=\{1,2,3,4\}\) है और \(A\times B\) में कुल (28) अवयव हैं, तो (|B|) क्या होगा?
If \(A=\{1,2,3,4\}\) and \(A\times B\) has (28) elements, what is (|B|)?
#unknown-cardinality
#formula
#sets
A (4)
B (7)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
Since (|A|=4) and \(4\cdot|B|=28\), we get (|B|=7). Divide to find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is B. (7). Since (|A|=4) and \(4\cdot|B|=28\), we get (|B|=7). Divide to find the unknown cardinality.
Step 3
Exam Tip
(|A|=4) और \(4\cdot|B|=28\), इसलिए (|B|=7)। अज्ञात कार्डिनलिटी के लिए भाग करें।
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यदि \(A=\{0,1,2\}\), \(B=\{1,2,3,4\}\) और \(C=\{2,4,6\}\) हैं, तो (|A\times\(B\cap C\)|) क्या होगा?
If \(A=\{0,1,2\}\), \(B=\{1,2,3,4\}\), and \(C=\{2,4,6\}\), what is (|A\times\(B\cap C\)|)?
#intersection
#cardinality
#set-operation
A (3)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={2,4}\), so the cardinality is \(3\cdot2=6\). Multiply only after finding the intersection.
Step 2
Why this answer is correct
The correct answer is B. (6). \(B\cap C={2,4}\), so the cardinality is \(3\cdot2=6\). Multiply only after finding the intersection.
Step 3
Exam Tip
\(B\cap C={2,4}\), इसलिए कार्डिनलिटी \(3\cdot2=6\) है। प्रतिच्छेद के बाद ही गुणा करें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) और \(C=\{4,6,8\}\) हैं, तो (|A\times\(B\cup C\)|) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\), and \(C=\{4,6,8\}\), what is (|A\times\(B\cup C\)|)?
#union
#cardinality
#cartesian-product
A (12)
B (14)
C (16)
D (20)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={2,4,6,8}\), so (|A\times\(B\cup C\)|=4\cdot4=16). Find the union first.
Step 2
Why this answer is correct
The correct answer is C. (16). \(B\cup C={2,4,6,8}\), so (|A\times\(B\cup C\)|=4\cdot4=16). Find the union first.
Step 3
Exam Tip
\(B\cup C={2,4,6,8}\), इसलिए (|A\times\(B\cup C\)|=4\cdot4=16)। पहले संघ निकालें।
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यदि (|A|=3) और (|B|=5) हैं, तो (A) से (B) तक संभव संबंधों की संख्या क्या है?
If (|A|=3) and (|B|=5), what is the number of possible relations from (A) to (B)?
#relations
#power-set
#cardinality
A (15)
B (125)
C (32768)
D (59049)
Explanation opens after your attempt
Correct Answer
C. (32768)
Step 1
Concept
A relation is a subset of \(A\times B\), and \(|A\times B|=15\), so the number is \(2^{15}=32768\). Number of relations uses a power of (2).
Step 2
Why this answer is correct
The correct answer is C. (32768). A relation is a subset of \(A\times B\), and \(|A\times B|=15\), so the number is \(2^{15}=32768\). Number of relations uses a power of (2).
Step 3
Exam Tip
संबंध \(A\times B\) का उपसमुच्चय है और \(|A\times B|=15\), इसलिए संख्या \(2^{15}=32768\) है। संबंधों की संख्या के लिए (2) की घात लगती है।
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यदि \(A=\{a,b\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(B\times A\) में कितने अवयव होंगे?
If \(A=\{a,b\}\) and \(B=\{1,2,3,4\}\), how many elements will \(B\times A\) have?
#reverse-product
#cardinality
#cartesian-product
A (6)
B (8)
C (10)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(|B\times A|=|B|\cdot|A|=4\cdot2=8\). Reversing order changes pairs but not the count.
Step 2
Why this answer is correct
The correct answer is B. (8). \(|B\times A|=|B|\cdot|A|=4\cdot2=8\). Reversing order changes pairs but not the count.
Step 3
Exam Tip
\(|B\times A|=|B|\cdot|A|=4\cdot2=8\)। क्रम बदलने पर युग्म बदलते हैं, पर संख्या समान रहती है।
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यदि \(A=\{1,2\}\), \(B=\{3,4,5\}\) और \(C=\{6,7\}\) हैं, तो \(|A\times B\times C|\) का मान क्या होगा?
If \(A=\{1,2\}\), \(B=\{3,4,5\}\), and \(C=\{6,7\}\), what is \(|A\times B\times C|\)?
#ordered-triple
#cardinality
#sets
A (7)
B (10)
C (12)
D (18)
Explanation opens after your attempt
Step 1
Concept
The cardinality for three sets is \(2\cdot3\cdot2=12\). For ordered triples, multiply all sizes.
Step 2
Why this answer is correct
The correct answer is C. (12). The cardinality for three sets is \(2\cdot3\cdot2=12\). For ordered triples, multiply all sizes.
Step 3
Exam Tip
तीन समुच्चयों में कार्डिनलिटी \(2\cdot3\cdot2=12\) होती है। क्रमित त्रिक में सभी आकार गुणा करें।
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यदि \(A={x:x\in\mathbb{N},3\le x\le8}\) और \(B={y:y\in\mathbb{N},y\le4}\) हैं, तो \(|A\times B|\) क्या है?
If \(A={x:x\in\mathbb{N},3\le x\le8}\) and \(B={y:y\in\mathbb{N},y\le4}\), what is \(|A\times B|\)?
#set-builder
#cardinality
#counting
A (18)
B (20)
C (24)
D (30)
Explanation opens after your attempt
Step 1
Concept
Here (|A|=6) and (|B|=4), so \(|A\times B|=6\cdot4=24\). Count the elements first.
Step 2
Why this answer is correct
The correct answer is C. (24). Here (|A|=6) and (|B|=4), so \(|A\times B|=6\cdot4=24\). Count the elements first.
Step 3
Exam Tip
यहाँ (|A|=6) और (|B|=4), इसलिए \(|A\times B|=6\cdot4=24\)। पहले समुच्चयों के अवयव गिनें।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{6,7\}\) हैं, तो (|\(A\times B\)\times C|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(C=\{6,7\}\), what is (|\(A\times B\)\times C|)?
#nested-product
#cardinality
#ordered-pair
A (7)
B (10)
C (12)
D (24)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=3\cdot2=6\), and then \(6\cdot2=12\). Cardinality still multiplies in nested products.
Step 2
Why this answer is correct
The correct answer is C. (12). \(|A\times B|=3\cdot2=6\), and then \(6\cdot2=12\). Cardinality still multiplies in nested products.
Step 3
Exam Tip
\(|A\times B|=3\cdot2=6\) और फिर \(6\cdot2=12\)। नेस्टेड गुणनफल में भी कार्डिनलिटी गुणा होती है।
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यदि \(A=\{1,2,3\}\) है और \(A\times B\) में कुल (15) अवयव हैं, तो (|B|) क्या होगा?
If \(A=\{1,2,3\}\) and \(A\times B\) has (15) elements, what is (|B|)?
#unknown-cardinality
#formula
#sets
A (3)
B (4)
C (5)
D (12)
Explanation opens after your attempt
Step 1
Concept
Since (|A|=3) and \(3\cdot|B|=15\), (|B|=5). Divide to find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is C. (5). Since (|A|=3) and \(3\cdot|B|=15\), (|B|=5). Divide to find the unknown cardinality.
Step 3
Exam Tip
(|A|=3) और \(3\cdot|B|=15\), इसलिए (|B|=5)। अज्ञात कार्डिनलिटी के लिए भाग करें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,5\}\) और \(C=\{2,4,5\}\) हैं, तो (|A\times\(B\cap C\)|) क्या होगा?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,5\}\), and \(C=\{2,4,5\}\), what is (|A\times\(B\cap C\)|)?
#intersection
#cardinality
#set-operation
A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={2,5}\), so the cardinality is \(4\cdot2=8\). Simplify the intersection first.
Step 2
Why this answer is correct
The correct answer is B. (8). \(B\cap C={2,5}\), so the cardinality is \(4\cdot2=8\). Simplify the intersection first.
Step 3
Exam Tip
\(B\cap C={2,5}\), इसलिए कार्डिनलिटी \(4\cdot2=8\) है। प्रतिच्छेद को पहले सरल करना जरूरी है।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{3,4,5\}\) हैं, तो (|A\times\(B\cup C\)|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), and \(C=\{3,4,5\}\), what is (|A\times\(B\cup C\)|)?
#union
#cardinality
#cartesian-product
A (9)
B (12)
C (15)
D (18)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={2,3,4,5}\), so (|A\times\(B\cup C\)|=3\cdot4=12). Find the union first, then multiply.
Step 2
Why this answer is correct
The correct answer is B. (12). \(B\cup C={2,3,4,5}\), so (|A\times\(B\cup C\)|=3\cdot4=12). Find the union first, then multiply.
Step 3
Exam Tip
\(B\cup C={2,3,4,5}\), इसलिए (|A\times\(B\cup C\)|=3\cdot4=12)। पहले संघ निकालें फिर गुणा करें।
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यदि (|A|=2) और (|B|=5) हैं, तो (A) से (B) तक संभव संबंधों की संख्या क्या है?
If (|A|=2) and (|B|=5), what is the number of possible relations from (A) to (B)?
#relations
#power-set
#cardinality
A (10)
B (25)
C (32)
D (1024)
Explanation opens after your attempt
Step 1
Concept
A relation is a subset of \(A\times B\), and \(|A\times B|=10\), so the number is \(2^{10}=1024\). Number of relations uses a power of (2).
Step 2
Why this answer is correct
The correct answer is D. (1024). A relation is a subset of \(A\times B\), and \(|A\times B|=10\), so the number is \(2^{10}=1024\). Number of relations uses a power of (2).
Step 3
Exam Tip
संबंध \(A\times B\) का उपसमुच्चय है और \(|A\times B|=10\), इसलिए संख्या \(2^{10}=1024\) है। संबंधों की संख्या के लिए घात (2) लगती है।
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यदि \(A=\{a,b,c\}\) और \(B=\{1,2\}\) हैं, तो \(B\times A\) में कितने अवयव होंगे?
If \(A=\{a,b,c\}\) and \(B=\{1,2\}\), how many elements will \(B\times A\) have?
#reverse-product
#cardinality
#sets
A (5)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(|B\times A|=|B|\cdot|A|=2\cdot3=6\). Reversing order changes pairs but not the cardinality.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|B\times A|=|B|\cdot|A|=2\cdot3=6\). Reversing order changes pairs but not the cardinality.
Step 3
Exam Tip
\(|B\times A|=|B|\cdot|A|=2\cdot3=6\)। क्रम बदलने से युग्म बदलते हैं, पर कार्डिनलिटी समान रहती है।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{6,7,8\}\) हैं, तो \(|A\times B\times C|\) का मान क्या होगा?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(C=\{6,7,8\}\), what is \(|A\times B\times C|\)?
#ordered-triple
#cardinality
#hard
A (8)
B (12)
C (18)
D (24)
Explanation opens after your attempt
Step 1
Concept
For product of three sets, the cardinality is \(3\cdot2\cdot3=18\). For ordered triples, multiply the sizes of all sets.
Step 2
Why this answer is correct
The correct answer is C. (18). For product of three sets, the cardinality is \(3\cdot2\cdot3=18\). For ordered triples, multiply the sizes of all sets.
Step 3
Exam Tip
तीन समुच्चयों के गुणनफल में कार्डिनलिटी \(3\cdot2\cdot3=18\) होती है। क्रमित त्रिक में सभी समुच्चयों के आकार गुणा करें।
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\(यदि (A={x:x\in\mathbb{N},2\le x\le6}) और (B={y:y\in\mathbb{N},y\) is odd\(,y<8}) हैं, तो (|A\times B|) क्या है\)?
\(If (A={x:x\in\mathbb{N},2\le x\le6}) and (B={y:y\in\mathbb{N},y\) is odd\(,y<8}), what is (|A\times B|)\)?
#set-builder
#cardinality
#cartesian-product
A (15)
B (20)
C (25)
D (30)
Explanation opens after your attempt
Step 1
Concept
Here \(A=\{2,3,4,5,6\}\) and \(B=\{1,3,5,7\}\), so \(|A\times B|=5\cdot4=20\). First count both sets.
Step 2
Why this answer is correct
The correct answer is B. (20). Here \(A=\{2,3,4,5,6\}\) and \(B=\{1,3,5,7\}\), so \(|A\times B|=5\cdot4=20\). First count both sets.
Step 3
Exam Tip
यहाँ \(A=\{2,3,4,5,6\}\) और \(B=\{1,3,5,7\}\), इसलिए \(|A\times B|=5\cdot4=20\)। पहले दोनों समुच्चयों की संख्या गिनें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{1,3,5\}\) हैं, तो (|\(A\cap B\)\times C|) क्या होगा?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{1,3,5\}\), what is (|\(A\cap B\)\times C|)?
#intersection
#cardinality
#set-operation
A (4)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B={2,3}\) and (|C|=3), so the cardinality is \(2\cdot3=6\). Simplify the intersection first.
Step 2
Why this answer is correct
The correct answer is C. (6). \(A\cap B={2,3}\) and (|C|=3), so the cardinality is \(2\cdot3=6\). Simplify the intersection first.
Step 3
Exam Tip
\(A\cap B={2,3}\) और (|C|=3), इसलिए कार्डिनलिटी \(2\cdot3=6\) है। प्रतिच्छेद को पहले सरल करें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{3,4,5\}\) हैं, तो (|A\times\(B\cap C\)|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{3,4,5\}\), what is (|A\times\(B\cap C\)|)?
#intersection
#cardinality
#cartesian-product
A (3)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={3,4}\), so (|A\times\(B\cap C\)|=3\cdot2=6). Multiply only after finding the intersection.
Step 2
Why this answer is correct
The correct answer is B. (6). \(B\cap C={3,4}\), so (|A\times\(B\cap C\)|=3\cdot2=6). Multiply only after finding the intersection.
Step 3
Exam Tip
\(B\cap C={3,4}\), इसलिए (|A\times\(B\cap C\)|=3\cdot2=6)। प्रतिच्छेद के बाद ही गुणा करें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5\}\) हैं, तो (A\times\(B\cup C\)) में कितने अवयव हैं?
If \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5\}\), how many elements are in (A\times\(B\cup C\))?
#union
#cartesian-product
#cardinality
A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={3,4,5}\), so (|A\times\(B\cup C\)|=2\cdot3=6). First evaluate the set inside the brackets.
Step 2
Why this answer is correct
The correct answer is C. (6). \(B\cup C={3,4,5}\), so (|A\times\(B\cup C\)|=2\cdot3=6). First evaluate the set inside the brackets.
Step 3
Exam Tip
\(B\cup C={3,4,5}\), इसलिए (|A\times\(B\cup C\)|=2\cdot3=6)। पहले कोष्ठक के भीतर का समुच्चय निकालें।
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यदि (|A|=2), (|B|=3) और (|C|=4), तो \(|A\times B\times C|\) क्या होगा?
If (|A|=2), (|B|=3), and (|C|=4), what is \(|A\times B\times C|\)?
#ordered-triple
#three-sets
#cardinality
A (9)
B (12)
C (24)
D \(2^9\)
Explanation opens after your attempt
Step 1
Concept
For Cartesian product of three sets, the cardinality is \(2\cdot3\cdot4=24\). For ordered triples, multiply all three sizes.
Step 2
Why this answer is correct
The correct answer is C. (24). For Cartesian product of three sets, the cardinality is \(2\cdot3\cdot4=24\). For ordered triples, multiply all three sizes.
Step 3
Exam Tip
तीन समुच्चयों के कार्तीय गुणनफल में कार्डिनलिटी \(2\cdot3\cdot4=24\) है। क्रमित त्रिक के प्रश्न में तीनों आकार गुणा करें।
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यदि (|A|=4), (|B|=5) और \(R\subseteq A\times B\), तो (R) में अधिकतम कितने अवयव हो सकते हैं?
If (|A|=4), (|B|=5), and \(R\subseteq A\times B\), what is the maximum possible number of elements in (R)?
#maximum-elements
#relation-subset
#cardinality
A (9)
B (20)
C (25)
D \(2^{20}\)
Explanation opens after your attempt
Step 1
Concept
Since (R) can be at most the whole \(A\times B\), and \(|A\times B|=20\). If maximum elements are asked, do not write \(2^{20}\).
Step 2
Why this answer is correct
The correct answer is B. (20). Since (R) can be at most the whole \(A\times B\), and \(|A\times B|=20\). If maximum elements are asked, do not write \(2^{20}\).
Step 3
Exam Tip
क्योंकि (R) अधिकतम पूरे \(A\times B\) के बराबर हो सकता है और \(|A\times B|=20\)। अधिकतम अवयव पूछे जाएँ तो \(2^{20}\) नहीं लिखें।
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यदि (A) में (m) अवयव और (B) में (n) अवयव हैं, तो \(A\times B\) के उपसमुच्चयों की संख्या क्या होगी?
If (A) has (m) elements and (B) has (n) elements, how many subsets does \(A\times B\) have?
#power-set
#cardinality
#general-form
A (mn)
B \(2^{m+n}\)
C \(2^{mn}\)
D \(m^n\)
Explanation opens after your attempt
Correct Answer
C. \(2^{mn}\)
Step 1
Concept
\(|A\times B|=mn\), so the number of subsets is \(2^{mn}\). First find the cardinality of the base set.
Step 2
Why this answer is correct
The correct answer is C. \(2^{mn}\). \(|A\times B|=mn\), so the number of subsets is \(2^{mn}\). First find the cardinality of the base set.
Step 3
Exam Tip
\(|A\times B|=mn\), इसलिए इसके उपसमुच्चयों की संख्या \(2^{mn}\) होगी। पहले मूल समुच्चय की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2,3\}\) है और \(|A\times B|=12\), तो (|B|) क्या होगा?
If \(A=\{1,2,3\}\) and \(|A\times B|=12\), what is (|B|)?
#unknown-cardinality
#formula
#sets
A (3)
B (4)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
Since (|A|=3) and \(3\cdot |B|=12\), we get (|B|=4). Divide to find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is B. (4). Since (|A|=3) and \(3\cdot |B|=12\), we get (|B|=4). Divide to find the unknown cardinality.
Step 3
Exam Tip
(|A|=3) और \(3\cdot |B|=12\), इसलिए (|B|=4)। अज्ञात कार्डिनलिटी निकालने में भाग दें।
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यदि \(A=\{1,2\}\), \(B=\{3\}\) और \(C=\{4,5,6\}\) हैं, तो (|\(A\times B\)\times C|) क्या होगा?
If \(A=\{1,2\}\), \(B=\{3\}\), and \(C=\{4,5,6\}\), what is (|\(A\times B\)\times C|)?
#nested-product
#cardinality
#hard
A (5)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=2\cdot1=2\), then (|\(A\times B\)\times C|=2\cdot3=6). Cardinalities multiply in nested products too.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|A\times B|=2\cdot1=2\), then (|\(A\times B\)\times C|=2\cdot3=6). Cardinalities multiply in nested products too.
Step 3
Exam Tip
\(|A\times B|=2\cdot1=2\) और फिर (|\(A\times B\)\times C|=2\cdot3=6)। संयुक्त गुणनफल में भी कार्डिनलिटी गुणा होती है।
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यदि \(A=\{0,1\}\) और \(B=\{2,4,6\}\) हैं, तो \(|A\times B|\) का मान क्या होगा?
If \(A=\{0,1\}\) and \(B=\{2,4,6\}\), what is the value of \(|A\times B|\)?
#cardinality
#cartesian-product
#counting
A (5)
B (6)
C (8)
D (3)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=|A|\cdot |B|=2\cdot3=6\). For cardinality questions, multiply the sizes directly.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|A\times B|=|A|\cdot |B|=2\cdot3=6\). For cardinality questions, multiply the sizes directly.
Step 3
Exam Tip
\(|A\times B|=|A|\cdot |B|=2\cdot3=6\) होता है। कार्डिनलिटी वाले प्रश्नों में सीधे गुणा करें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{3,4,5\}\) हैं, तो (A\times(\(B\cup C\)-A)) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) and \(C=\{3,4,5\}\), how many elements are in (A\times(\(B\cup C\)-A))?
#cartesian-product
#union
#set-difference
#cardinality
A (3)
B (4)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={2,3,4,5}\) and (\(B\cup C\)-A={4,5}). Therefore (n(A\times(\(B\cup C\)-A))=3\times2=6).
Step 2
Why this answer is correct
The correct answer is C. (6). \(B\cup C={2,3,4,5}\) and (\(B\cup C\)-A={4,5}). Therefore (n(A\times(\(B\cup C\)-A))=3\times2=6).
Step 3
Exam Tip
\(B\cup C={2,3,4,5}\) और (\(B\cup C\)-A={4,5}) है। इसलिए (n(A\times(\(B\cup C\)-A))=3\times2=6)।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{6\}\) हैं, तो (\(A\times B\)\times C) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\) and \(C=\{6\}\), how many elements are in (\(A\times B\)\times C)?
#cartesian-product
#nested-product
#cardinality
A (5)
B (6)
C (12)
D (3)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=3\times2=6) and (n(C)=1). Therefore (n(\(A\times B\)\times C)=6\times1=6).
Step 2
Why this answer is correct
The correct answer is B. (6). (n\(A\times B\)=3\times2=6) and (n(C)=1). Therefore (n(\(A\times B\)\times C)=6\times1=6).
Step 3
Exam Tip
(n\(A\times B\)=3\times2=6) और (n(C)=1) है। इसलिए (n(\(A\times B\)\times C)=6\times1=6)।
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यदि \(A=\{m,n,o\}\) है, तो \(A\times A\) में कितने क्रमित युग्म होंगे?
If \(A=\{m,n,o\}\), how many ordered pairs are in \(A\times A\)?
#cartesian-product
#self-product
#cardinality
A (6)
B (3)
C (12)
D (9)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times A\)=n(A)2 =32 =9). Pairs with equal components are also included.
Step 2
Why this answer is correct
The correct answer is D. (9). (n\(A\times A\)=n(A)2 =32 =9). Pairs with equal components are also included.
Step 3
Exam Tip
(n\(A\times A\)=n(A)2 =32 =9) होता है। समान घटक वाले युग्म भी इसमें शामिल होते हैं।
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यदि (n(A)=6) और (n\(A\times B\)=48) है, तो (n(B)) कितना होगा?
If (n(A)=6) and (n\(A\times B\)=48), what is (n(B))?
#cartesian-product
#cardinality
#numerical
A (6)
B (42)
C (8)
D (54)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)), so (48=6n(B)) and (n(B)=8). Use the formula in reverse in such questions.
Step 2
Why this answer is correct
The correct answer is C. (8). (n\(A\times B\)=n(A)n(B)), so (48=6n(B)) and (n(B)=8). Use the formula in reverse in such questions.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)), इसलिए (48=6n(B)) और (n(B)=8)। ऐसे प्रश्नों में सूत्र को उल्टा लगाएं।
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यदि \(A=\{2,4,6,8\}\), \(B=\{4,8\}\) और \(C=\{1,3,5\}\) हैं, तो \((A-B)\times C\) में कितने अवयव होंगे?
If \(A=\{2,4,6,8\}\), \(B=\{4,8\}\) and \(C=\{1,3,5\}\), how many elements are in \((A-B)\times C\)?
#cartesian-product
#set-difference
#cardinality
A (9)
B (6)
C (5)
D (12)
Explanation opens after your attempt
Step 1
Concept
(A-B={2,6}), so (n\((A-B)\times C\)=2\times3=6). It is necessary to find the set difference first.
Step 2
Why this answer is correct
The correct answer is B. (6). (A-B={2,6}), so (n\((A-B)\times C\)=2\times3=6). It is necessary to find the set difference first.
Step 3
Exam Tip
(A-B={2,6}), इसलिए (n\((A-B)\times C\)=2\times3=6)। पहले समुच्चय अंतर निकालना जरूरी है।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{1,3,5\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) and \(C=\{1,3,5\}\), how many elements are in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
A (6)
B (9)
C (3)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={3}\), so (n(A\times\(B\cap C\))=3\times1=3). Find the intersection first and then multiply.
Step 2
Why this answer is correct
The correct answer is C. (3). \(B\cap C={3}\), so (n(A\times\(B\cap C\))=3\times1=3). Find the intersection first and then multiply.
Step 3
Exam Tip
\(B\cap C={3}\), इसलिए (n(A\times\(B\cap C\))=3\times1=3)। पहले प्रतिच्छेद निकालें फिर गुणन करें।
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यदि \(A=\{1,4,7\}\) और \(B=\{0,2\}\) हैं, तो \(A\times B\) में कुल कितने क्रमित युग्म होंगे?
If \(A=\{1,4,7\}\) and \(B=\{0,2\}\), how many ordered pairs are there in \(A\times B\)?
#cartesian-product
#cardinality
#ordered-pairs
A (5)
B (6)
C (3)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)=3\times2=6). Counting elements of both sets first is the safest method.
Step 2
Why this answer is correct
The correct answer is B. (6). (n\(A\times B\)=n(A)n(B)=3\times2=6). Counting elements of both sets first is the safest method.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)=3\times2=6) होता है। पहले दोनों समुच्चयों के अवयव गिनना सबसे सुरक्षित तरीका है।
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यदि \(A=\{1,2,5\}\), \(B=\{2,3,5\}\) और \(C=\{5,7\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,5\}\), \(B=\{2,3,5\}\) and \(C=\{5,7\}\), how many elements are there in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
A (3)
B (6)
C (9)
D (1)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={5}\), so (n(A\times\(B\cap C\))=3\times1=3). First find the intersection and then count the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. (3). \(B\cap C={5}\), so (n(A\times\(B\cap C\))=3\times1=3). First find the intersection and then count the Cartesian product.
Step 3
Exam Tip
\(B\cap C={5}\), इसलिए (n(A\times\(B\cap C\))=3\times1=3)। पहले प्रतिच्छेद निकालें फिर कार्तीय गुणन गिनें।
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यदि \(A=\{1,2\}\), \(B=\{3\}\) और \(C=\{4,5\}\) हैं, तो (A\times\(B\times C\)) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{3\}\) and \(C=\{4,5\}\), how many elements are in (A\times\(B\times C\))?
#cartesian-product
#nested-product
#cardinality
A (4)
B (5)
C (6)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(B\times C\)=1\times2=2), so (n(A\times\(B\times C\))=2\times2=4). Count the inner Cartesian product first.
Step 2
Why this answer is correct
The correct answer is A. (4). (n\(B\times C\)=1\times2=2), so (n(A\times\(B\times C\))=2\times2=4). Count the inner Cartesian product first.
Step 3
Exam Tip
(n\(B\times C\)=1\times2=2), इसलिए (n(A\times\(B\times C\))=2\times2=4)। अंदर का कार्तीय गुणन पहले गिनें।
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यदि \(A=\{p,q\}\) है, तो \(A\times A\) में कितने क्रमित युग्म होंगे?
If \(A=\{p,q\}\), how many ordered pairs are in \(A\times A\)?
#cartesian-product
#self-product
#cardinality
A (4)
B (2)
C (3)
D (1)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times A\)=n(A)2 =22 =4). Pairs with equal components such as ((p,p)) and ((q,q)) are also included.
Step 2
Why this answer is correct
The correct answer is A. (4). (n\(A\times A\)=n(A)2 =22 =4). Pairs with equal components such as ((p,p)) and ((q,q)) are also included.
Step 3
Exam Tip
(n\(A\times A\)=n(A)2 =22 =4) होता है। ((p,p)) और ((q,q)) जैसे समान घटक वाले युग्म भी शामिल होते हैं।
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यदि (n(A)=5) और (n\(A\times B\)=35) है, तो (n(B)) कितना होगा?
If (n(A)=5) and (n\(A\times B\)=35), what is (n(B))?
#cartesian-product
#cardinality
#numerical
A (7)
B (30)
C (40)
D (5)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)), so (35=5n(B)) and (n(B)=7). In such questions, use the formula in reverse too.
Step 2
Why this answer is correct
The correct answer is A. (7). (n\(A\times B\)=n(A)n(B)), so (35=5n(B)) and (n(B)=7). In such questions, use the formula in reverse too.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)), इसलिए (35=5n(B)) और (n(B)=7)। ऐसे प्रश्नों में सूत्र को उल्टा भी लगाएं।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,3,5\}\) और \(C=\{0,2\}\) हैं, तो \((A-B)\times C\) में कितने अवयव होंगे?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,3,5\}\) and \(C=\{0,2\}\), how many elements are there in \((A-B)\times C\)?
#cartesian-product
#set-difference
#cardinality
A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
(A-B={2,4}), so (n\((A-B)\times C\)=2\times2=4). It is necessary to find the set difference first.
Step 2
Why this answer is correct
The correct answer is A. (4). (A-B={2,4}), so (n\((A-B)\times C\)=2\times2=4). It is necessary to find the set difference first.
Step 3
Exam Tip
(A-B={2,4}), इसलिए (n\((A-B)\times C\)=2\times2=4)। पहले समुच्चय अंतर निकालना जरूरी है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) और \(C=\{4,6,8\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) and \(C=\{4,6,8\}\), how many elements are in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
A (8)
B (12)
C (6)
D (4)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={4,6}\), so (n(A\times\(B\cap C\))=4\times2=8). First find the intersection and then multiply.
Step 2
Why this answer is correct
The correct answer is A. (8). \(B\cap C={4,6}\), so (n(A\times\(B\cap C\))=4\times2=8). First find the intersection and then multiply.
Step 3
Exam Tip
\(B\cap C={4,6}\), इसलिए (n(A\times\(B\cap C\))=4\times2=8)। पहले प्रतिच्छेद निकालें फिर गुणन करें।
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यदि \(A=\{2,4,6\}\) और \(B=\{1,3\}\) हैं, तो \(A\times B\) में कुल कितने क्रमित युग्म होंगे?
If \(A=\{2,4,6\}\) and \(B=\{1,3\}\), how many ordered pairs are there in \(A\times B\)?
#cartesian-product
#cardinality
#ordered-pairs
A (6)
B (5)
C (3)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)=3\times2=6). In exams, first count the elements of both sets.
Step 2
Why this answer is correct
The correct answer is A. (6). (n\(A\times B\)=n(A)n(B)=3\times2=6). In exams, first count the elements of both sets.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)=3\times2=6) होता है। परीक्षा में पहले दोनों समुच्चयों के अवयव गिनें।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{2,4\}\) और \(C=\{0,1,2\}\) हैं, तो \((A-B)\times C\) में कितने अवयव होंगे?
If \(A=\{1,2,3,4,5\}\), \(B=\{2,4\}\) and \(C=\{0,1,2\}\), how many elements are there in \((A-B)\times C\)?
#cartesian-product
#set-difference
#cardinality
A (9)
B (6)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
(A-B={1,3,5}), so (n\((A-B)\times C\)=3\times3=9). First find the set difference and then count the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. (9). (A-B={1,3,5}), so (n\((A-B)\times C\)=3\times3=9). First find the set difference and then count the Cartesian product.
Step 3
Exam Tip
(A-B={1,3,5}), इसलिए (n\((A-B)\times C\)=3\times3=9)। पहले समुच्चय का अंतर निकालें फिर कार्तीय गुणन गिनें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{3,4,5\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) and \(C=\{3,4,5\}\), how many elements are in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
A (6)
B (9)
C (3)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={3,4}\), so (n(A\times\(B\cap C\))=3\times2=6). Find the intersection before counting the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. (6). \(B\cap C={3,4}\), so (n(A\times\(B\cap C\))=3\times2=6). Find the intersection before counting the Cartesian product.
Step 3
Exam Tip
\(B\cap C={3,4}\), इसलिए (n(A\times\(B\cap C\))=3\times2=6)। प्रतिच्छेद निकालकर ही कार्तीय गुणन गिनें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5,6\}\) हैं, तो (A\times\(B\cup C\)) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(C=\{5,6\}\), how many elements are in (A\times\(B\cup C\))?
#cartesian-product
#union
#cardinality
A (8)
B (6)
C (4)
D (10)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={3,4,5,6}\), so (n(A\times\(B\cup C\))=2\times4=8). Complete the set operation first.
Step 2
Why this answer is correct
The correct answer is A. (8). \(B\cup C={3,4,5,6}\), so (n(A\times\(B\cup C\))=2\times4=8). Complete the set operation first.
Step 3
Exam Tip
\(B\cup C={3,4,5,6}\), इसलिए (n(A\times\(B\cup C\))=2\times4=8)। पहले समुच्चय संक्रिया पूरी करें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5\}\) हैं, तो (\(A\times B\)\times C) में अवयवों की संख्या क्या होगी?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(C=\{5\}\), what is the number of elements in (\(A\times B\)\times C)?
#cartesian-product
#nested-product
#cardinality
A (4)
B (5)
C (8)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=4) and (n(C)=1), so (n(\(A\times B\)\times C)=4). Remember that (\(A\times B\)) itself is a set.
Step 2
Why this answer is correct
The correct answer is A. (4). (n\(A\times B\)=4) and (n(C)=1), so (n(\(A\times B\)\times C)=4). Remember that (\(A\times B\)) itself is a set.
Step 3
Exam Tip
(n\(A\times B\)=4) और (n(C)=1), इसलिए (n(\(A\times B\)\times C)=4)। ध्यान रखें कि (\(A\times B\)) खुद एक समुच्चय है।
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यदि \(A=\{p,q,r\}\) है और \(A\times A\) बनाया जाता है, तो इसमें कितने क्रमित युग्म होंगे?
If \(A=\{p,q,r\}\) and \(A\times A\) is formed, how many ordered pairs will it contain?
#cartesian-product
#self-product
#cardinality
A (9)
B (6)
C (3)
D (12)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times A\)=n(A)2 =32 =9). In \(A\times A\), pairs with equal elements are also included.
Step 2
Why this answer is correct
The correct answer is A. (9). (n\(A\times A\)=n(A)2 =32 =9). In \(A\times A\), pairs with equal elements are also included.
Step 3
Exam Tip
(n\(A\times A\)=n(A)2 =32 =9) होता है। \(A\times A\) में समान अवयव वाले युग्म भी शामिल होते हैं।
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यदि (n(A)=4) और (n\(A\times B\)=20) है, तो (n(B)) कितना होगा?
If (n(A)=4) and (n\(A\times B\)=20), what is (n(B))?
#cartesian-product
#cardinality
#numerical
A (5)
B (16)
C (24)
D (80)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)), so (20=4n(B)) and (n(B)=5). Use the multiplication formula directly in such questions.
Step 2
Why this answer is correct
The correct answer is A. (5). (n\(A\times B\)=n(A)n(B)), so (20=4n(B)) and (n(B)=5). Use the multiplication formula directly in such questions.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)), इसलिए (20=4n(B)) और (n(B)=5)। ऐसे प्रश्नों में गुणन सूत्र सीधे लगाएं।
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यदि \(A=\{1,2,3\}\) और \(B=\varnothing\) हैं, तो \(A\times B\) क्या होगा?
If \(A=\{1,2,3\}\) and \(B=\varnothing\), what is \(A\times B\)?
#cartesian-product
#empty-set
#cardinality
A \(\varnothing\)
B ({1,2,3})
C ({(1,0),(2,0),(3,0)})
D ({\(\varnothing,1\)})
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
The Cartesian product of any set with an empty set is \(\varnothing\). If one set is empty, no ordered pair is formed.
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). The Cartesian product of any set with an empty set is \(\varnothing\). If one set is empty, no ordered pair is formed.
Step 3
Exam Tip
किसी भी समुच्चय का रिक्त समुच्चय के साथ कार्तीय गुणन \(\varnothing\) होता है। यदि एक भी समुच्चय रिक्त हो तो कोई क्रमित युग्म नहीं बनता।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) में कितने क्रमित युग्म होंगे?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), how many ordered pairs are there in \(A\times B\)?
#cartesian-product
#cardinality
#ordered-pairs
A (6)
B (5)
C (3)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)=2\times3=6). In exams, first count elements of both sets.
Step 2
Why this answer is correct
The correct answer is A. (6). (n\(A\times B\)=n(A)n(B)=2\times3=6). In exams, first count elements of both sets.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)=2\times3=6) होता है। परीक्षा में पहले दोनों समुच्चयों के अवयव गिनें।
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यदि \(A={x:x\in \mathbb{N},\ x\leq 3}\) और \(B=\{0,1\}\) है, तो \(A\times B\) में कितने तत्व हैं?
If \(A={x:x\in \mathbb{N},\ x\leq 3}\) and \(B=\{0,1\}\), how many elements are in \(A\times B\)?
#cartesian product
#set builder
#cardinality
#level24
A (3)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(A=\{1,2,3\}\) and (B) has (2) elements, so \(3\times 2=6\). First understand the set-builder form.
Step 2
Why this answer is correct
The correct answer is C. (6). \(A=\{1,2,3\}\) and (B) has (2) elements, so \(3\times 2=6\). First understand the set-builder form.
Step 3
Exam Tip
\(A=\{1,2,3\}\) और (B) में (2) तत्व हैं, इसलिए \(3\times 2=6\)। पहले सेट-बिल्डर रूप को समझें।
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यदि \(M=\{1,2,3\}\) और \(N=\{4\}\) हैं, तो \(M\times N\) और \(N\times M\) के बारे में सही कथन कौन सा है?
If \(M=\{1,2,3\}\) and \(N=\{4\}\), which statement about \(M\times N\) and \(N\times M\) is correct?
#cartesian product
#reversal
#cardinality
#level24
A दोनों में (3) युग्म हैं पर युग्मों का क्रम अलग है / Both have (3) pairs but the order of components is different
B दोनों बिल्कुल समान हैं / Both are exactly equal
C दोनों खाली हैं / Both are empty
D \(M\times N\) में (1) और \(N\times M\) में (3) युग्म हैं / \(M\times N\) has (1) pair and \(N\times M\) has (3) pairs
Explanation opens after your attempt
Correct Answer
A. दोनों में (3) युग्म हैं पर युग्मों का क्रम अलग है / Both have (3) pairs but the order of components is different
Step 1
Concept
Both have count \(3\times 1=3\), but the order of components is reversed. Equal count does not mean equal sets.
Step 2
Why this answer is correct
The correct answer is A. दोनों में (3) युग्म हैं पर युग्मों का क्रम अलग है / Both have (3) pairs but the order of components is different. Both have count \(3\times 1=3\), but the order of components is reversed. Equal count does not mean equal sets.
Step 3
Exam Tip
दोनों की संख्या \(3\times 1=3\) है, पर घटकों का क्रम उल्टा है। समान संख्या का अर्थ समान समुच्चय नहीं होता।
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यदि \(C=\{2,3,5\}\) है, तो \(C\times C\) में कितने तत्व होंगे?
If \(C=\{2,3,5\}\), how many elements are in \(C\times C\)?
#cartesian product
#self product
#cardinality
#level24
A (3)
B (6)
C (9)
D (15)
Explanation opens after your attempt
Step 1
Concept
(n(C)=3), so (n\(C\times C\)=3\times 3=9). The multiplication rule applies even with the same set.
Step 2
Why this answer is correct
The correct answer is C. (9). (n(C)=3), so (n\(C\times C\)=3\times 3=9). The multiplication rule applies even with the same set.
Step 3
Exam Tip
(n(C)=3), इसलिए (n\(C\times C\)=3\times 3=9)। समान समुच्चय के साथ भी गुणा नियम लागू रहता है।
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यदि \(A=\{0,2,4\}\) और \(B=\varnothing\) हैं, तो (n\(A\times B\)) कितना है?
If \(A=\{0,2,4\}\) and \(B=\varnothing\), what is (n\(A\times B\))?
#cartesian product
#empty set
#cardinality
#level24
A (0)
B (3)
C (4)
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
(B) is empty, so no second component is available and no pair is formed. Since count is asked, write (0).
Step 2
Why this answer is correct
The correct answer is A. (0). (B) is empty, so no second component is available and no pair is formed. Since count is asked, write (0).
Step 3
Exam Tip
(B) खाली है, इसलिए कोई दूसरा घटक नहीं मिलेगा और कोई युग्म नहीं बनेगा। संख्या पूछी गई है, इसलिए (0) लिखें।
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यदि (n\(A\times B\)=24) और (n(A)=6) है, तो (n(B)) कितना होगा?
If (n\(A\times B\)=24) and (n(A)=6), what is (n(B))?
#cartesian product
#unknown cardinality
#numerical
#level24
A (3)
B (4)
C (6)
D (18)
Explanation opens after your attempt
Step 1
Concept
\(24=6\times n(B)\), so (n(B)=4). For an unknown count, divide total pairs by the known count.
Step 2
Why this answer is correct
The correct answer is B. (4). \(24=6\times n(B)\), so (n(B)=4). For an unknown count, divide total pairs by the known count.
Step 3
Exam Tip
\(24=6\times n(B)\), इसलिए (n(B)=4)। अज्ञात संख्या के लिए कुल युग्मों को ज्ञात संख्या से भाग दें।
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यदि \(A=\{m,n\}\) और \(B=\{p,q,r\}\) हैं, तो (n\(B\times A\)) कितना है?
If \(A=\{m,n\}\) and \(B=\{p,q,r\}\), what is (n\(B\times A\))?
#cartesian product
#reversed product
#cardinality
#level24
A (2)
B (3)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
(B) has (3) elements and (A) has (2) elements, so \(3\times 2=6\). Reversing order changes the pairs, not the count.
Step 2
Why this answer is correct
The correct answer is D. (6). (B) has (3) elements and (A) has (2) elements, so \(3\times 2=6\). Reversing order changes the pairs, not the count.
Step 3
Exam Tip
(B) में (3) और (A) में (2) तत्व हैं, इसलिए \(3\times 2=6\)। क्रम बदलने से संख्या नहीं, केवल युग्मों का क्रम बदलता है।
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यदि (n(A)=7) और (n(B)=3) है, तो (n\(A\times B\)) कितना होगा?
If (n(A)=7) and (n(B)=3), what is (n\(A\times B\))?
#cartesian product
#cardinality
#formula
#level24
A (10)
B (21)
C (7)
D (3)
Explanation opens after your attempt
Step 1
Concept
For finite sets, (n\(A\times B\)=n(A)n(B)). Therefore, \(7\times 3=21\).
Step 2
Why this answer is correct
The correct answer is B. (21). For finite sets, (n\(A\times B\)=n(A)n(B)). Therefore, \(7\times 3=21\).
Step 3
Exam Tip
सीमित समुच्चयों के लिए (n\(A\times B\)=n(A)n(B))। इसलिए \(7\times 3=21\)।
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यदि \(P=\{3,6,9\}\) और \(Q=\{0,1\}\) हैं, तो (n\(P\times Q\)) कितना होगा?
If \(P=\{3,6,9\}\) and \(Q=\{0,1\}\), what is (n\(P\times Q\))?
#cartesian product
#cardinality
#counting
#level24
A (3)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
(n\(P\times Q\)=n(P)n(Q)=3\times 2=6). To find the number in Cartesian product, multiply.
Step 2
Why this answer is correct
The correct answer is C. (6). (n\(P\times Q\)=n(P)n(Q)=3\times 2=6). To find the number in Cartesian product, multiply.
Step 3
Exam Tip
(n\(P\times Q\)=n(P)n(Q)=3\times 2=6)। कार्तीय गुणन में संख्या निकालने के लिए गुणा करें।
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यदि \(A={x:x\in \mathbb{N},\ x\leq 2}\) और \(B=\{10,20\}\) है, तो \(A\times B\) में कितने तत्व हैं?
If \(A={x:x\in \mathbb{N},\ x\leq 2}\) and \(B=\{10,20\}\), how many elements are in \(A\times B\)?
#cartesian product
#set builder
#cardinality
#level23
A (2)
B (4)
C (10)
D (20)
Explanation opens after your attempt
Step 1
Concept
\(A=\{1,2\}\) and (B) has (2) elements, so \(2\times 2=4\). First convert set-builder form into a list.
Step 2
Why this answer is correct
The correct answer is B. (4). \(A=\{1,2\}\) and (B) has (2) elements, so \(2\times 2=4\). First convert set-builder form into a list.
Step 3
Exam Tip
\(A=\{1,2\}\) और (B) में (2) तत्व हैं, इसलिए \(2\times 2=4\)। पहले सेट-बिल्डर रूप को सूची में बदलें।
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यदि \(M=\{2,5\}\) और \(N=\{8,9\}\) हैं, तो \(M\times N\) और \(N\times M\) के बारे में सही कथन कौन सा है?
If \(M=\{2,5\}\) and \(N=\{8,9\}\), which statement about \(M\times N\) and \(N\times M\) is correct?
#cartesian product
#reversal
#cardinality
#level23
A दोनों की संख्या बराबर है पर युग्मों का क्रम अलग है / Both have equal count but the order of pairs is different
B दोनों हमेशा बिल्कुल समान हैं / Both are always exactly equal
C दोनों खाली हैं / Both are empty
D \(M\times N\) में (2) और \(N\times M\) में (4) युग्म हैं / \(M\times N\) has (2) pairs and \(N\times M\) has (4) pairs
Explanation opens after your attempt
Correct Answer
A. दोनों की संख्या बराबर है पर युग्मों का क्रम अलग है / Both have equal count but the order of pairs is different
Step 1
Concept
Both have \(2\times 2=4\) pairs, but the order of components differs. Equal count does not mean the sets are necessarily equal.
Step 2
Why this answer is correct
The correct answer is A. दोनों की संख्या बराबर है पर युग्मों का क्रम अलग है / Both have equal count but the order of pairs is different. Both have \(2\times 2=4\) pairs, but the order of components differs. Equal count does not mean the sets are necessarily equal.
Step 3
Exam Tip
दोनों में \(2\times 2=4\) युग्म होंगे, पर घटकों का क्रम अलग होगा। संख्या बराबर होने से समुच्चय समान होना जरूरी नहीं है।
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यदि \(A=\{1,2,3,4\}\) है, तो (n\(A\times A\)) कितना होगा?
If \(A=\{1,2,3,4\}\), what is (n\(A\times A\))?
#cartesian product
#self product
#cardinality
#level23
A (4)
B (8)
C (16)
D (20)
Explanation opens after your attempt
Step 1
Concept
(n(A)=4), so (n\(A\times A\)=4\times 4=16). With the same set, the count behaves like a square.
Step 2
Why this answer is correct
The correct answer is C. (16). (n(A)=4), so (n\(A\times A\)=4\times 4=16). With the same set, the count behaves like a square.
Step 3
Exam Tip
(n(A)=4), इसलिए (n\(A\times A\)=4\times 4=16)। समान समुच्चय होने पर भी संख्या वर्ग की तरह आती है।
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यदि \(A=\varnothing\) और \(B=\{9\}\) हैं, तो (n\(A\times B\)) कितना है?
If \(A=\varnothing\) and \(B=\{9\}\), what is (n\(A\times B\))?
#cartesian product
#empty set
#cardinality
#level23
A (0)
B (1)
C (9)
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
There is no element in (A) for the first component, so no pair is formed. Since the number is asked, the answer is (0).
Step 2
Why this answer is correct
The correct answer is A. (0). There is no element in (A) for the first component, so no pair is formed. Since the number is asked, the answer is (0).
Step 3
Exam Tip
पहले घटक के लिए (A) में कोई तत्व नहीं है, इसलिए कोई युग्म नहीं बनेगा। संख्या पूछी गई है, इसलिए उत्तर (0) है।
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यदि (n\(A\times B\)=15) और (n(A)=5) है, तो (n(B)) कितना होगा?
If (n\(A\times B\)=15) and (n(A)=5), what is (n(B))?
#cartesian product
#unknown cardinality
#formula
#level23
A (2)
B (3)
C (5)
D (10)
Explanation opens after your attempt
Step 1
Concept
\(15=5\times n(B)\), so (n(B)=3). To find the unknown count, divide the total pairs by the known count.
Step 2
Why this answer is correct
The correct answer is B. (3). \(15=5\times n(B)\), so (n(B)=3). To find the unknown count, divide the total pairs by the known count.
Step 3
Exam Tip
\(15=5\times n(B)\), इसलिए (n(B)=3)। अज्ञात संख्या निकालने के लिए कुल युग्मों को ज्ञात संख्या से भाग दें।
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यदि (n(A)=6) और (n(B)=2) है, तो (n\(A\times B\)) कितना होगा?
If (n(A)=6) and (n(B)=2), what is (n\(A\times B\))?
#cartesian product
#formula
#cardinality
#level23
A (8)
B (12)
C (6)
D (4)
Explanation opens after your attempt
Step 1
Concept
For finite sets, (n\(A\times B\)=n(A)n(B)). Therefore, \(6\times 2=12\).
Step 2
Why this answer is correct
The correct answer is B. (12). For finite sets, (n\(A\times B\)=n(A)n(B)). Therefore, \(6\times 2=12\).
Step 3
Exam Tip
सीमित समुच्चयों के लिए (n\(A\times B\)=n(A)n(B))। इसलिए \(6\times 2=12\)।
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यदि \(P=\{0,4,8\}\) और \(Q=\{1\}\) हैं, तो \(P\times Q\) में कितने क्रमित युग्म होंगे?
If \(P=\{0,4,8\}\) and \(Q=\{1\}\), how many ordered pairs are in \(P\times Q\)?
#cartesian product
#cardinality
#counting
#level23
A (1)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
(n\(P\times Q\)=n(P)n(Q)=3\times 1=3). In counting questions, multiply the number of elements.
Step 2
Why this answer is correct
The correct answer is B. (3). (n\(P\times Q\)=n(P)n(Q)=3\times 1=3). In counting questions, multiply the number of elements.
Step 3
Exam Tip
(n\(P\times Q\)=n(P)n(Q)=3\times 1=3)। गिनती वाले प्रश्न में तत्वों की संख्या गुणा करें।
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यदि (n\(A\times B\)=8) और (n(B)=2) है, तो (n(A)) कितना होगा?
If (n\(A\times B\)=8) and (n(B)=2), what is (n(A))?
#cartesian product
#unknown cardinality
#formula
#level22
A (2)
B (4)
C (6)
D (10)
Explanation opens after your attempt
Step 1
Concept
\(8=n(A)\times 2\), so (n(A)=4). For an unknown count, divide the total pairs by the known count.
Step 2
Why this answer is correct
The correct answer is B. (4). \(8=n(A)\times 2\), so (n(A)=4). For an unknown count, divide the total pairs by the known count.
Step 3
Exam Tip
\(8=n(A)\times 2\), इसलिए (n(A)=4)। अज्ञात संख्या के लिए कुल युग्मों को ज्ञात संख्या से भाग दें।
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यदि \(X=\{2,4\}\) और \(Y=\{1,3,5\}\) है, तो (n\(Y\times X\)) कितना होगा?
If \(X=\{2,4\}\) and \(Y=\{1,3,5\}\), what is (n\(Y\times X\))?
#cartesian product
#cardinality
#reversed notation
#level22
A (5)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
(n(Y)=3) and (n(X)=2), so (n\(Y\times X\)=3\times 2=6). The same multiplication rule applies even when set names change.
Step 2
Why this answer is correct
The correct answer is B. (6). (n(Y)=3) and (n(X)=2), so (n\(Y\times X\)=3\times 2=6). The same multiplication rule applies even when set names change.
Step 3
Exam Tip
(n(Y)=3) और (n(X)=2), इसलिए (n\(Y\times X\)=3\times 2=6)। नाम बदलने पर भी वही गुणा नियम लागू होता है।
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यदि (n(A)=m) और (n(B)=n) है, तो (n\(A\times B\)) का सूत्र क्या है?
If (n(A)=m) and (n(B)=n), what is the formula for (n\(A\times B\))?
#cartesian product
#formula
#cardinality
#level22
A (m+n)
B (mn)
C (m-n)
D \(\frac{m}{n}\)
Explanation opens after your attempt
Step 1
Concept
In Cartesian product, each element of (A) pairs with every element of (B), so the total count is (mn). For counting questions, multiply, do not add.
Step 2
Why this answer is correct
The correct answer is B. (mn). In Cartesian product, each element of (A) pairs with every element of (B), so the total count is (mn). For counting questions, multiply, do not add.
Step 3
Exam Tip
कार्तीय गुणन में प्रत्येक (A) तत्व (B) के हर तत्व से जुड़ता है, इसलिए कुल संख्या (mn) होती है। गिनती वाले प्रश्न में जोड़ नहीं, गुणा करें।
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सीमित समुच्चयों के लिए (n\(A\times B\)) और (n\(B\times A\)) के बारे में सही कथन कौन सा है?
For finite sets, which statement about (n\(A\times B\)) and (n\(B\times A\)) is correct?
#cartesian product
#cardinality
#reversal
#level22
A (n\(A\times B\)=n\(B\times A\))
B (n\(A\times B\)>n\(B\times A\)) हमेशा / (n\(A\times B\)>n\(B\times A\)) always
C (n\(A\times B\)<n\(B\times A\)) हमेशा / (n\(A\times B\)<n\(B\times A\)) always
D (n\(B\times A\)=0) हमेशा / (n\(B\times A\)=0) always
Explanation opens after your attempt
Correct Answer
A. (n\(A\times B\)=n\(B\times A\))
Step 1
Concept
Both counts are (n(A)n(B)), so the numbers are equal. Remember that counts may be equal even when the sets are different.
Step 2
Why this answer is correct
The correct answer is A. (n\(A\times B\)=n\(B\times A\)). Both counts are (n(A)n(B)), so the numbers are equal. Remember that counts may be equal even when the sets are different.
Step 3
Exam Tip
दोनों की संख्या (n(A)n(B)) होती है, इसलिए संख्याएं बराबर होती हैं। ध्यान रखें कि संख्या बराबर हो सकती है पर समुच्चय अलग हो सकते हैं।
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यदि \(A=\{9\}\) और \(B=\varnothing\) है, तो (n\(A\times B\)) कितना होगा?
If \(A=\{9\}\) and \(B=\varnothing\), what is (n\(A\times B\))?
#cartesian product
#empty set
#cardinality
#level22
A (0)
B (1)
C (9)
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
There is no element in (B), so no second component is available. Since the number is asked, the answer is (0).
Step 2
Why this answer is correct
The correct answer is A. (0). There is no element in (B), so no second component is available. Since the number is asked, the answer is (0).
Step 3
Exam Tip
(B) में कोई तत्व नहीं है, इसलिए कोई दूसरा घटक नहीं मिल सकता। संख्या पूछी गई है, इसलिए उत्तर (0) है।
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यदि (n\(A\times B\)=12) और (n(A)=3) है, तो (n(B)) कितना होगा?
If (n\(A\times B\)=12) and (n(A)=3), what is (n(B))?
#cartesian product
#unknown cardinality
#numerical
#level22
A (3)
B (4)
C (9)
D (15)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)), so (12=3n(B)) and (n(B)=4). Divide to find the unknown count.
Step 2
Why this answer is correct
The correct answer is B. (4). (n\(A\times B\)=n(A)n(B)), so (12=3n(B)) and (n(B)=4). Divide to find the unknown count.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)), इसलिए (12=3n(B)) और (n(B)=4)। अज्ञात संख्या निकालने के लिए भाग दें।
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