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यदि (n(A)=3), (n(B)=4) और (A,B) दोनों अरिक्त हैं, तो \(A\times B\) और \(B\times A\) में अवयवों की संख्या का संबंध क्या है?
If (n(A)=3), (n(B)=4) and both (A,B) are non-empty, what is the relation between the numbers of elements in \(A\times B\) and \(B\times A\)?
#cartesian-product
#comparison
#noncommutative
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A दोनों में (12) अवयव होंगे / both have (12) elements
B \(A\times B\) में (7) अवयव होंगे / \(A\times B\) has (7) elements
C \(B\times A\) में (1) अवयव होगा / \(B\times A\) has (1) element
D दोनों हमेशा समान समुच्चय होंगे / both are always equal sets
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Correct Answer
A. दोनों में (12) अवयव होंगे / both have (12) elements
Step 1
Concept
Both have \(3\times4=12\) elements. The number can be the same, but the order of pairs is generally different.
Step 2
Why this answer is correct
The correct answer is A. दोनों में (12) अवयव होंगे / both have (12) elements. Both have \(3\times4=12\) elements. The number can be the same, but the order of pairs is generally different.
Step 3
Exam Tip
दोनों में अवयवों की संख्या \(3\times4=12\) होती है। संख्या समान हो सकती है पर युग्मों का क्रम सामान्यतः अलग होता है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो (A) से (B) तक कुल कितने संबंध संभव हैं?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), how many relations are possible from (A) to (B)?
#cartesian-product
#relations
#subsets
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A \(2^6\)
B (6)
C \(3^2\)
D \(2^3\)
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Correct Answer
A. \(2^6\)
Step 1
Concept
(n\(A\times B\)=3\times2=6), and every relation is a subset of \(A\times B\). Therefore the total number of relations is \(2^6\).
Step 2
Why this answer is correct
The correct answer is A. \(2^6\). (n\(A\times B\)=3\times2=6), and every relation is a subset of \(A\times B\). Therefore the total number of relations is \(2^6\).
Step 3
Exam Tip
(n\(A\times B\)=3\times2=6), और हर संबंध \(A\times B\) का उपसमुच्चय होता है। इसलिए कुल संबंध \(2^6\) हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो \(A\times B\) का कौन सा उपसमुच्चय (A) से (B) तक संबंध है?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), which subset of \(A\times B\) is a relation from (A) to (B)?
#cartesian-product
#relation
#subset
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A ({(1,4),(3,5)})
B ({(4,1),(5,3)})
C ({(1,6)})
D ({(0,4),(2,5)})
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Correct Answer
A. ({(1,4),(3,5)})
Step 1
Concept
A relation is a subset of \(A\times B\). Only all pairs of ({(1,4),(3,5)}) belong to \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. ({(1,4),(3,5)}). A relation is a subset of \(A\times B\). Only all pairs of ({(1,4),(3,5)}) belong to \(A\times B\).
Step 3
Exam Tip
संबंध \(A\times B\) का उपसमुच्चय होता है। केवल ({(1,4),(3,5)}) के सभी युग्म \(A\times B\) में हैं।
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\(यदि (A={1,2,3}) और (B={a,b}) हैं, तो (A\times B) में कौन सा युग्म उस संबंध (R={(x,y):x\in A,,y\in B,,x\) विषम है}) में होगा?
\(If (A={1,2,3}) and (B={a,b}), which pair belongs to the relation (R={(x,y):x\in A,,y\in B,,x\) is odd})?
#cartesian-product
#relation
#condition
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A ((3,b))
B ((2,a))
C ((a,3))
D ((b,1))
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Correct Answer
A. ((3,b))
Step 1
Concept
\(3\in A\) is odd and \(b\in B\), so \((3,b)\in R\). Apply the given condition to the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. ((3,b)). \(3\in A\) is odd and \(b\in B\), so \((3,b)\in R\). Apply the given condition to the Cartesian product.
Step 3
Exam Tip
\(3\in A\) विषम है और \(b\in B\), इसलिए \((3,b)\in R\)। संबंध में दी गई शर्त को कार्तीय गुणन पर लागू करें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(R=\{(1,3),(2,4)\}\) है, तो (R) किसका उपसमुच्चय है?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(R=\{(1,3),(2,4)\}\), then (R) is a subset of which set?
#cartesian-product
#subset
#relation
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A \(A\times B\)
B \(B\times A\)
C \(A\cap B\)
D \(A\cup B\)
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Correct Answer
A. \(A\times B\)
Step 1
Concept
In every pair of (R), the first component is from (A) and the second is from (B). Therefore \(R\subseteq A\times B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\times B\). In every pair of (R), the first component is from (A) and the second is from (B). Therefore \(R\subseteq A\times B\).
Step 3
Exam Tip
(R) के हर युग्म में पहला घटक (A) से और दूसरा (B) से है। इसलिए \(R\subseteq A\times B\)।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कौन सा युग्म उस संबंध \(R=\{(x,y):y=2x\}\) में नहीं होगा?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6\}\), which pair in \(A\times B\) will not belong to the relation \(R=\{(x,y):y=2x\}\)?
#cartesian-product
#relation
#condition
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A ((1,4))
B ((1,2))
C ((2,4))
D ((3,6))
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Correct Answer
A. ((1,4))
Step 1
Concept
In ((1,4)), \(4\ne2\times1\), so it is not in (R). Apply the given relation condition to each option.
Step 2
Why this answer is correct
The correct answer is A. ((1,4)). In ((1,4)), \(4\ne2\times1\), so it is not in (R). Apply the given relation condition to each option.
Step 3
Exam Tip
((1,4)) में \(4\ne2\times1\), इसलिए यह (R) में नहीं है। संबंध में दी गई शर्त को हर विकल्प पर लगाएं।
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