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यदि \(R\subseteq A\times B\) और \(R=\{(1,4),(2,4),(2,5),(3,6)\}\) है, तो (R) का domain क्या है?
If \(R\subseteq A\times B\) and \(R=\{(1,4),(2,4),(2,5),(3,6)\}\), what is the domain of (R)?
#domain
#relations
#ordered-pairs
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A ({4,5,6})
B ({1,2,3})
C ({1,2,4,5})
D ({2,4})
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Correct Answer
B. ({1,2,3})
Step 1
Concept
The domain is the set of first components. Here the first components are (1,2,3).
Step 2
Why this answer is correct
The correct answer is B. ({1,2,3}). The domain is the set of first components. Here the first components are (1,2,3).
Step 3
Exam Tip
domain पहले घटकों का समुच्चय होता है। यहां पहले घटक (1,2,3) हैं।
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यदि \(R=\{(1,4),(2,4),(2,5),(3,6)\}\) है, तो (R) का range क्या है?
If \(R=\{(1,4),(2,4),(2,5),(3,6)\}\), what is the range of (R)?
#range
#relations
#ordered-pairs
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A ({1,2,3})
B ({4,5,6})
C ({2,4,6})
D ({1,4})
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Correct Answer
B. ({4,5,6})
Step 1
Concept
The range is the set of second components. The repeated (4) is written only once.
Step 2
Why this answer is correct
The correct answer is B. ({4,5,6}). The range is the set of second components. The repeated (4) is written only once.
Step 3
Exam Tip
range दूसरे घटकों का समुच्चय होता है। दोहराए गए (4) को एक बार ही लिखा जाता है।
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यदि \(R=\{(x,y)\in \mathbb{R}\times\mathbb{R}:x^2=y^2\}\), तो (R) के लिए सही कथन क्या है?
If \(R=\{(x,y)\in \mathbb{R}\times\mathbb{R}:x^2=y^2\}\), which statement is correct for (R)?
#equivalence-relation
#real-numbers
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A समतुल्य संबंध है / It is an equivalence relation
B केवल सममित है / It is only symmetric
C प्रतिवर्ती नहीं है / It is not reflexive
D संक्रमी नहीं है / It is not transitive
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Correct Answer
A. समतुल्य संबंध है / It is an equivalence relation
Step 1
Concept
\(x^2=x^2\), \(x^2=y^2\Rightarrow y^2=x^2\), and equality is transitive. Hence it is an equivalence relation.
Step 2
Why this answer is correct
The correct answer is A. समतुल्य संबंध है / It is an equivalence relation. \(x^2=x^2\), \(x^2=y^2\Rightarrow y^2=x^2\), and equality is transitive. Hence it is an equivalence relation.
Step 3
Exam Tip
\(x^2=x^2\), \(x^2=y^2\Rightarrow y^2=x^2\), और बराबरी की शर्त संक्रमी है। इसलिए यह समतुल्य संबंध है।
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यदि (R) एक संबंध है और (\operatorname{Dom}(R)={2,3,5}), (\operatorname{Ran}(R)={7,11}), तो (R) कम से कम कितने युग्मों वाला हो सकता है?
If (R) is a relation with (\operatorname{Dom}(R)={2,3,5}) and (\operatorname{Ran}(R)={7,11}), what is the minimum possible number of pairs in (R)?
#domain
#range
#minimal-relation
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A (2)
B (3)
C (5)
D (6)
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Step 1
Concept
All three domain elements must appear in some pair, so at least (3) pairs are needed. The (2) range elements can be covered within these (3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). All three domain elements must appear in some pair, so at least (3) pairs are needed. The (2) range elements can be covered within these (3) pairs.
Step 3
Exam Tip
तीनों domain अवयवों को किसी न किसी युग्म में आना होगा, इसलिए कम से कम (3) युग्म चाहिए। range के (2) अवयव इन्हीं (3) युग्मों में कवर किए जा सकते हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{x,y\}\), और \(R=\{(1,x),(2,x),(3,y)\}\), तो (R) का range क्या है?
If \(A=\{1,2,3\}\), \(B=\{x,y\}\), and \(R=\{(1,x),(2,x),(3,y)\}\), what is the range of (R)?
#range
#relations
#sets
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A ({1,2,3})
B ({x,y})
C ({x})
D ({y})
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Correct Answer
B. ({x,y})
Step 1
Concept
Both second components (x) and (y) occur. Therefore the range is ({x,y}).
Step 2
Why this answer is correct
The correct answer is B. ({x,y}). Both second components (x) and (y) occur. Therefore the range is ({x,y}).
Step 3
Exam Tip
दूसरे घटक (x) और (y) दोनों आते हैं। इसलिए range ({x,y}) है।
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समुच्चय \(A=\{1,2,3,4\}\) पर संबंध \(R=\{(a,b):a+b=6\}\) का domain क्या है?
For the relation \(R=\{(a,b):a+b=6\}\) on \(A=\{1,2,3,4\}\), what is the domain?
#domain
#sum-relation
#ordered-pairs
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A ({1,2,3,4})
B ({2,3,4})
C ({1,2,3})
D ({2,4})
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Correct Answer
B. ({2,3,4})
Step 1
Concept
The valid pairs are ((2,4),(3,3),(4,2)). Their first components are ({2,3,4}).
Step 2
Why this answer is correct
The correct answer is B. ({2,3,4}). The valid pairs are ((2,4),(3,3),(4,2)). Their first components are ({2,3,4}).
Step 3
Exam Tip
मान्य युग्म ((2,4),(3,3),(4,2)) हैं। इनके पहले घटक ({2,3,4}) हैं।
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\(A=\{1,2,3,4\}\) पर \(R=\{(a,b):a+b=7\}\) का range क्या है?
On \(A=\{1,2,3,4\}\), what is the range of \(R=\{(a,b):a+b=7\}\)?
#range
#sum-relation
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A ({1,2,3,4})
B ({3,4})
C ({2,3,4})
D ({1,4})
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Correct Answer
B. ({3,4})
Step 1
Concept
The valid pairs are ((3,4)) and ((4,3)). Hence the set of second components is ({3,4}).
Step 2
Why this answer is correct
The correct answer is B. ({3,4}). The valid pairs are ((3,4)) and ((4,3)). Hence the set of second components is ({3,4}).
Step 3
Exam Tip
मान्य युग्म ((3,4)) और ((4,3)) हैं। इसलिए दूसरे घटकों का समुच्चय ({3,4}) है।
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