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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
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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 \(A=\{1,2,3\}\), \(B=\{1,2,3\}\) और \(R=\{(a,b)\in A\times B:a+b=4\}\), तो (R) क्या है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3\}\), and \(R=\{(a,b)\in A\times B:a+b=4\}\), what is (R)?
#cartesian-product
#relation-subset
#sum-condition
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A ({(1,3),(2,2),(3,1)})
B ({(1,2),(2,1)})
C ({(3,3)})
D ({(1,1),(2,2),(3,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(2,2),(3,1)})
Step 1
Concept
All ordered pairs with sum (4) are ((1,3),(2,2),(3,1)). A relation is a subset of \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,2),(3,1)}). All ordered pairs with sum (4) are ((1,3),(2,2),(3,1)). A relation is a subset of \(A\times B\).
Step 3
Exam Tip
योग (4) देने वाले सभी क्रमित युग्म ((1,3),(2,2),(3,1)) हैं। संबंध \(A\times B\) का उपसमुच्चय होता है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक (2) क्रमित युग्म हों?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many subsets of \(A\times B\) contain exactly (2) ordered pairs?
#cartesian-product
#relations
#counting-combinations
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A (16)
B (64)
C (120)
D (256)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=16), so the number of ways to choose exactly (2) pairs is \({}^{16}C_2=120\). Use subset selection when counting relations.
Step 2
Why this answer is correct
The correct answer is C. (120). (n\(A\times B\)=16), so the number of ways to choose exactly (2) pairs is \({}^{16}C_2=120\). Use subset selection when counting relations.
Step 3
Exam Tip
(n\(A\times B\)=16), इसलिए ठीक (2) युग्म चुनने के तरीके \({}^{16}C_2=120\) हैं। संबंध गिनते समय उपसमुच्चय चयन का विचार लगाएं।
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यदि (n(A)=3) और (n(B)=2), तो (A) से (B) तक कुल कितने संबंध संभव हैं?
If (n(A)=3) and (n(B)=2), how many relations from (A) to (B) are possible?
#cartesian-product
#relations
#total-relations
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A (6)
B (12)
C (32)
D (64)
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Step 1
Concept
\(A\times B\) has \(3\cdot2=6\) pairs, and each relation is a subset of it. Hence total relations are \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is D. (64). \(A\times B\) has \(3\cdot2=6\) pairs, and each relation is a subset of it. Hence total relations are \(2^6=64\).
Step 3
Exam Tip
\(A\times B\) में \(3\cdot2=6\) युग्म हैं और हर संबंध इसका उपसमुच्चय है। इसलिए कुल संबंध \(2^6=64\) हैं।
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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
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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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