यदि \(A=\{1,2\}\) और \(B=\{3,4\}\) हैं, तो (A) से (B) में संबंध किसका उपसमुच्चय होता है?
If \(A=\{1,2\}\) and \(B=\{3,4\}\), then a relation from (A) to (B) is a subset of what?
#relations
#cartesian product
#definition
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A \(A\cup B\)
B \(A\cap B\)
C \(A\times B\)
D \(B\times A\)
Explanation opens after your attempt
Correct Answer
C. \(A\times B\)
Step 1
Concept
A relation from (A) to (B) is always a subset of \(A\times B\). In exams first identify the Cartesian product.
Step 2
Why this answer is correct
The correct answer is C. \(A\times B\). A relation from (A) to (B) is always a subset of \(A\times B\). In exams first identify the Cartesian product.
Step 3
Exam Tip
(A) से (B) में संबंध हमेशा \(A\times B\) का उपसमुच्चय होता है। परीक्षा में पहले कार्तीय गुणनफल पहचानें।
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यदि \(A=\{1,2,3\}\) और \(R=\{(x,y):x<y\}\), जहां \(x,y\in A\), तो (R) का सही रूप क्या है?
If \(A=\{1,2,3\}\) and \(R=\{(x,y):x<y\}\), where \(x,y\in A\), what is the correct form of (R)?
#relations
#inequality
#set builder
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A ({(1,2),(1,3),(2,3)})
B ({(2,1),(3,1),(3,2)})
C ({(1,1),(2,2),(3,3)})
D ({(1,2),(2,1)})
Explanation opens after your attempt
Correct Answer
A. ({(1,2),(1,3),(2,3)})
Step 1
Concept
The pairs satisfying (x<y) are ((1,2),(1,3),(2,3)). In exams, check all possible ordered pairs systematically.
Step 2
Why this answer is correct
The correct answer is A. ({(1,2),(1,3),(2,3)}). The pairs satisfying (x<y) are ((1,2),(1,3),(2,3)). In exams, check all possible ordered pairs systematically.
Step 3
Exam Tip
शर्त (x<y) को पूरा करने वाले युग्म ((1,2),(1,3),(2,3)) हैं। परीक्षा में सभी संभावित युग्मों को क्रम से जांचें।
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यदि \(A=\{a,b\}\) और \(B=\{1\}\) हैं, तो \(A\times B\) में कितने क्रमित युग्म होंगे?
If \(A=\{a,b\}\) and \(B=\{1\}\), how many ordered pairs are in \(A\times B\)?
#relations
#counting
#ordered pairs
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
Here \(|A\times B|=|A|\cdot |B|=2\cdot1=2\). This formula is very useful for counting relations.
Step 2
Why this answer is correct
The correct answer is B. (2). Here \(|A\times B|=|A|\cdot |B|=2\cdot1=2\). This formula is very useful for counting relations.
Step 3
Exam Tip
\(|A\times B|=|A|\cdot |B|=2\cdot1=2\)। संबंधों में गिनती के लिए यह सूत्र बहुत उपयोगी है।
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यदि \(R=\{(2,1),(3,1),(3,2)\}\) है, तो (R) का प्रतिलोम संबंध \(R^{-1}\) क्या होगा?
If \(R=\{(2,1),(3,1),(3,2)\}\), what will be the inverse relation \(R^{-1}\)?
#inverse relation
#ordered pairs
#relations
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A ({(1,2),(1,3),(2,3)})
B ({(2,1),(3,1),(3,2)})
C ({(1,1),(2,2),(3,3)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({(1,2),(1,3),(2,3)})
Step 1
Concept
In an inverse relation, the order of components in every pair is reversed. So ((2,1)) becomes ((1,2)), and similarly for the other pairs.
Step 2
Why this answer is correct
The correct answer is A. ({(1,2),(1,3),(2,3)}). In an inverse relation, the order of components in every pair is reversed. So ((2,1)) becomes ((1,2)), and similarly for the other pairs.
Step 3
Exam Tip
प्रतिलोम संबंध में हर युग्म के अवयवों का क्रम बदल जाता है। इसलिए ((2,1)) से ((1,2)) और इसी तरह अन्य युग्म मिलते हैं।
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समुच्चय \(A=\{1,2\}\) पर कुल कितने संबंध बनाए जा सकते हैं?
How many total relations can be formed on the set \(A=\{1,2\}\)?
#relations
#total relations
#power set
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
A relation on (A) is a subset of \(A\times A\), and \(|A\times A|=4\), so total relations are \(2^4=16\). Remember subset counting for exams.
Step 2
Why this answer is correct
The correct answer is D. (16). A relation on (A) is a subset of \(A\times A\), and \(|A\times A|=4\), so total relations are \(2^4=16\). Remember subset counting for exams.
Step 3
Exam Tip
(A) पर संबंध \(A\times A\) का उपसमुच्चय होता है और \(|A\times A|=4\), इसलिए कुल संबंध \(2^4=16\) हैं। परीक्षा में उपसमुच्चयों की संख्या याद रखें।
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संबंध \(R=\{(1,2),(2,3),(3,4)\}\) का प्रांत क्या है?
What is the domain of the relation \(R=\{(1,2),(2,3),(3,4)\}\)?
#domain
#relations
#ordered pairs
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A ({1,2,3})
B ({2,3,4})
C ({1,2,3,4})
D ({(1,2),(2,3),(3,4)})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3})
Step 1
Concept
The domain is the set of first components of ordered pairs. Here the first components are (1,2,3).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3}). The domain is the set of first components of ordered pairs. Here the first components are (1,2,3).
Step 3
Exam Tip
प्रांत क्रमित युग्मों के पहले अवयवों का समुच्चय होता है। यहां पहले अवयव (1,2,3) हैं।
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संबंध \(R=\{(2,5),(3,5),(4,6)\}\) का परिसर क्या है?
What is the range of the relation \(R=\{(2,5),(3,5),(4,6)\}\)?
#range
#relations
#basic
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A ({2,3,4})
B ({5,6})
C ({2,3,4,5,6})
D ({(2,5),(3,5)})
Explanation opens after your attempt
Correct Answer
B. ({5,6})
Step 1
Concept
The range is the set of second components, and repeated values are written once. So the range is ({5,6}).
Step 2
Why this answer is correct
The correct answer is B. ({5,6}). The range is the set of second components, and repeated values are written once. So the range is ({5,6}).
Step 3
Exam Tip
परिसर क्रमित युग्मों के दूसरे अवयवों का समुच्चय होता है और दोहराव केवल एक बार लिखा जाता है। इसलिए परिसर ({5,6}) है।
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यदि संबंध (R) में \((3,7)\in R\) है, तो इस युग्म में पहला अवयव क्या कहलाता है?
If \((3,7)\in R\), what is the first component of this ordered pair called?
#preimage
#image
#relations
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A प्रतिबिंब / Image
B पूर्व प्रतिबिंब / Preimage
C परिसर / Range
D सहप्रांत / Codomain
Explanation opens after your attempt
Correct Answer
B. पूर्व प्रतिबिंब / Preimage
Step 1
Concept
In the ordered pair ((3,7)), (3) is the preimage and (7) is the image. While reading relations, the first component forms the domain.
Step 2
Why this answer is correct
The correct answer is B. पूर्व प्रतिबिंब / Preimage. In the ordered pair ((3,7)), (3) is the preimage and (7) is the image. While reading relations, the first component forms the domain.
Step 3
Exam Tip
क्रमित युग्म ((3,7)) में (3) पूर्व प्रतिबिंब और (7) प्रतिबिंब माना जाता है। संबंध पढ़ते समय पहले अवयव से प्रांत बनता है।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(R=\{(1,4),(3,5)\}\) है, तो (R) का सहप्रांत क्या है?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(R=\{(1,4),(3,5)\}\), what is the codomain of (R)?
#codomain
#range
#relations
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A ({1,3})
B ({4,5})
C ({1,2,3})
D ({(1,4),(3,5)})
Explanation opens after your attempt
Correct Answer
B. ({4,5})
Step 1
Concept
For a relation from (A) to (B), the codomain is the whole set (B). The range may be a subset of the codomain.
Step 2
Why this answer is correct
The correct answer is B. ({4,5}). For a relation from (A) to (B), the codomain is the whole set (B). The range may be a subset of the codomain.
Step 3
Exam Tip
(A) से (B) में संबंध का सहप्रांत पूरा समुच्चय (B) होता है। परिसर सहप्रांत का उपसमुच्चय हो सकता है।
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यदि \(A=\{1,2\}\) और \(B=\{x,y\}\) हैं, तो खाली संबंध कौन-सा है?
If \(A=\{1,2\}\) and \(B=\{x,y\}\), which one is the empty relation?
#empty relation
#subset
#relations
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A \(R=\varnothing\)
B \(R=A\times B\)
C \(R=\{(1,x)\}\)
D \(R=\{(2,y)\}\)
Explanation opens after your attempt
Correct Answer
A. \(R=\varnothing\)
Step 1
Concept
An empty relation has no ordered pair, so \(R=\varnothing\). It is also a subset of \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. \(R=\varnothing\). An empty relation has no ordered pair, so \(R=\varnothing\). It is also a subset of \(A\times B\).
Step 3
Exam Tip
खाली संबंध में कोई क्रमित युग्म नहीं होता, इसलिए \(R=\varnothing\) है। यह भी \(A\times B\) का उपसमुच्चय होता है।
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यदि \(A=\{p,q\}\) है, तो (A) पर सार्वत्रिक संबंध कौन-सा है?
If \(A=\{p,q\}\), which is the universal relation on (A)?
#universal relation
#relations
#set
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A \(\varnothing\)
B ({(p,p),(q,q)})
C \(A\times A\)
D ({(p,q)})
Explanation opens after your attempt
Correct Answer
C. \(A\times A\)
Step 1
Concept
A universal relation contains all ordered pairs of \(A\times A\). Therefore the answer is \(A\times A\).
Step 2
Why this answer is correct
The correct answer is C. \(A\times A\). A universal relation contains all ordered pairs of \(A\times A\). Therefore the answer is \(A\times A\).
Step 3
Exam Tip
सार्वत्रिक संबंध में \(A\times A\) के सभी क्रमित युग्म शामिल होते हैं। इसलिए उत्तर \(A\times A\) है।
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समुच्चय \(A=\{1,2,3\}\) पर तत्समक संबंध कौन-सा है?
Which is the identity relation on \(A=\{1,2,3\}\)?
#identity relation
#relations
#basic
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A ({(1,2),(2,3),(3,1)})
B ({(1,1),(2,2),(3,3)})
C ({(1,1),(1,2),(1,3)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
B. ({(1,1),(2,2),(3,3)})
Step 1
Concept
In an identity relation, for every \(a\in A\), the pair ((a,a)) is included. So pairs with equal components are selected.
Step 2
Why this answer is correct
The correct answer is B. ({(1,1),(2,2),(3,3)}). In an identity relation, for every \(a\in A\), the pair ((a,a)) is included. So pairs with equal components are selected.
Step 3
Exam Tip
तत्समक संबंध में हर \(a\in A\) के लिए केवल ((a,a)) होता है। इसलिए सभी समान अवयवों वाले युग्म लिए जाते हैं।
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यदि \(R=\{(1,2),(2,4),(3,6)\}\) है, तो \(R^{-1}\) क्या होगा?
If \(R=\{(1,2),(2,4),(3,6)\}\), what is \(R^{-1}\)?
#inverse relation
#ordered pairs
#relations
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A ({(2,1),(4,2),(6,3)})
B ({(1,2),(2,4),(3,6)})
C ({(2,4),(4,6)})
D ({(1,3),(2,4),(3,6)})
Explanation opens after your attempt
Correct Answer
A. ({(2,1),(4,2),(6,3)})
Step 1
Concept
In the inverse relation, the two components of each ordered pair are interchanged. Thus ((1,2)) becomes ((2,1)).
Step 2
Why this answer is correct
The correct answer is A. ({(2,1),(4,2),(6,3)}). In the inverse relation, the two components of each ordered pair are interchanged. Thus ((1,2)) becomes ((2,1)).
Step 3
Exam Tip
प्रतिलोम संबंध में हर क्रमित युग्म के दोनों अवयव बदल दिए जाते हैं। इसलिए ((1,2)) से ((2,1)) बनता है।
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यदि \(A=\{1,2,3\}\) और \(R=\{(a,b):a=b\}\) है, जहां \(a,b\in A\), तो (R) किस प्रकार का संबंध है?
If \(A=\{1,2,3\}\) and \(R=\{(a,b):a=b\}\), where \(a,b\in A\), what type of relation is (R)?
#identity relation
#set builder
#relations
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A खाली संबंध / Empty relation
B तत्समक संबंध / Identity relation
C सार्वत्रिक संबंध / Universal relation
D केवल एक युग्म वाला संबंध / Single-pair relation
Explanation opens after your attempt
Correct Answer
B. तत्समक संबंध / Identity relation
Step 1
Concept
The condition (a=b) gives only ((1,1),(2,2),(3,3)). This is the definition of identity relation.
Step 2
Why this answer is correct
The correct answer is B. तत्समक संबंध / Identity relation. The condition (a=b) gives only ((1,1),(2,2),(3,3)). This is the definition of identity relation.
Step 3
Exam Tip
शर्त (a=b) केवल ((1,1),(2,2),(3,3)) देती है। यह तत्समक संबंध की परिभाषा है।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4\}\), \(R=\{(a,b):a<b\}\) है, तो कौन-सा युग्म (R) में है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), and \(R=\{(a,b):a<b\}\), which pair belongs to (R)?
#set builder
#relations
#inequality
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A ((3,2))
B ((2,2))
C ((1,4))
D ((4,1))
Explanation opens after your attempt
Correct Answer
C. ((1,4))
Step 1
Concept
In ((1,4)), (1<4) is true and both components are from the correct sets. Check both the condition and the sets in a relation.
Step 2
Why this answer is correct
The correct answer is C. ((1,4)). In ((1,4)), (1<4) is true and both components are from the correct sets. Check both the condition and the sets in a relation.
Step 3
Exam Tip
((1,4)) में (1<4) सत्य है और दोनों अवयव सही समुच्चयों से हैं। संबंध में शर्त और समुच्चय दोनों जांचें।
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यदि \(A=\{1,2,3\}\) पर \(R=\{(1,1),(2,2),(3,3),(1,2)\}\) है, तो (R) में कौन-सा अतिरिक्त युग्म तत्समक संबंध से अलग है?
If \(R=\{(1,1),(2,2),(3,3),(1,2)\}\) on \(A=\{1,2,3\}\), which extra pair is different from the identity relation?
#identity relation
#ordered pairs
#common mistake
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A ((1,1))
B ((2,2))
C ((3,3))
D ((1,2))
Explanation opens after your attempt
Correct Answer
D. ((1,2))
Step 1
Concept
An identity relation contains only pairs like ((a,a)). In ((1,2)), the two components are not equal.
Step 2
Why this answer is correct
The correct answer is D. ((1,2)). An identity relation contains only pairs like ((a,a)). In ((1,2)), the two components are not equal.
Step 3
Exam Tip
तत्समक संबंध में केवल ((a,a)) जैसे युग्म होते हैं। ((1,2)) में दोनों अवयव समान नहीं हैं।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो (A) से (B) में कुल कितने संबंध संभव हैं?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), how many relations are possible from (A) to (B)?
#counting relations
#cartesian product
#relations
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A \(2^5\)
B \(2^6\)
C (6)
D (12)
Explanation opens after your attempt
Correct Answer
B. \(2^6\)
Step 1
Concept
Here \(|A\times B|=2\cdot3=6\), so the number of relations is \(2^6\). Every subset forms a relation.
Step 2
Why this answer is correct
The correct answer is B. \(2^6\). Here \(|A\times B|=2\cdot3=6\), so the number of relations is \(2^6\). Every subset forms a relation.
Step 3
Exam Tip
\(|A\times B|=2\cdot3=6\), इसलिए संबंधों की संख्या \(2^6\) है। हर उपसमुच्चय एक संबंध बनाता है।
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यदि किसी संबंध का प्रांत ({1,2}) और परिसर ({5}) है, तो कौन-सा संबंध सही हो सकता है?
If a relation has domain ({1,2}) and range ({5}), which relation can be correct?
#domain
#range
#relations
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A ({(1,5),(2,5)})
B ({(1,5),(2,6)})
C ({(1,6),(2,6)})
D ({(5,1),(5,2)})
Explanation opens after your attempt
Correct Answer
A. ({(1,5),(2,5)})
Step 1
Concept
The first components give domain ({1,2}), and the second components give range ({5}). Therefore the first option is correct.
Step 2
Why this answer is correct
The correct answer is A. ({(1,5),(2,5)}). The first components give domain ({1,2}), and the second components give range ({5}). Therefore the first option is correct.
Step 3
Exam Tip
पहले अवयवों से प्रांत ({1,2}) और दूसरे अवयवों से परिसर ({5}) मिलता है। इसलिए पहला विकल्प सही है।
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संबंध \(R=\{(x,y):x+y=5\}\), जहां \(x,y\in{1,2,3,4}\), में कौन-सा युग्म नहीं है?
In the relation \(R=\{(x,y):x+y=5\}\), where \(x,y\in{1,2,3,4}\), which pair is not in (R)?
#set builder
#relations
#checking condition
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A ((1,4))
B ((2,3))
C ((3,2))
D ((4,2))
Explanation opens after your attempt
Correct Answer
D. ((4,2))
Step 1
Concept
For ((4,2)), (4+2=6), which is not (5). In such questions substitute each option in the condition.
Step 2
Why this answer is correct
The correct answer is D. ((4,2)). For ((4,2)), (4+2=6), which is not (5). In such questions substitute each option in the condition.
Step 3
Exam Tip
((4,2)) के लिए (4+2=6), जो (5) नहीं है। ऐसे प्रश्नों में हर विकल्प को शर्त में रखें।
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यदि \(R=\{(1,3),(2,4)\}\) \(A=\{1,2\}\) से \(B=\{3,4,5\}\) में है, तो (R) का परिसर क्या है?
If \(R=\{(1,3),(2,4)\}\) is a relation from \(A=\{1,2\}\) to \(B=\{3,4,5\}\), what is the range of (R)?
#range
#codomain
#relations
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A ({1,2})
B ({3,4})
C ({3,4,5})
D ({1,2,3,4})
Explanation opens after your attempt
Correct Answer
B. ({3,4})
Step 1
Concept
The range is formed only from second components actually appearing in the relation. (5) is in the codomain, but not in the range.
Step 2
Why this answer is correct
The correct answer is B. ({3,4}). The range is formed only from second components actually appearing in the relation. (5) is in the codomain, but not in the range.
Step 3
Exam Tip
परिसर केवल उन दूसरे अवयवों से बनता है जो संबंध में सच में आए हैं। (5) सहप्रांत में है, पर परिसर में नहीं है।
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यदि \(A=\{1,2,3\}\) है, तो \(A\times A\) में कितने युग्म होंगे?
If \(A=\{1,2,3\}\), how many pairs will be in \(A\times A\)?
#cartesian product
#relations
#counting
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A (3)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
Here \(|A\times A|=|A|^2=3^2=9\). This value is needed before counting relations.
Step 2
Why this answer is correct
The correct answer is C. (9). Here \(|A\times A|=|A|^2=3^2=9\). This value is needed before counting relations.
Step 3
Exam Tip
\(|A\times A|=|A|^2=3^2=9\)। संबंधों की संख्या निकालने से पहले यही मान चाहिए।
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यदि \(A=\{1,2,3\}\) पर संबंध \(R=\{(1,1),(2,2),(3,3)\}\) है, तो (R) का प्रांत क्या है?
If \(R=\{(1,1),(2,2),(3,3)\}\) is a relation on \(A=\{1,2,3\}\), what is the domain of (R)?
#domain
#identity relation
#relations
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A ({1})
B ({1,2,3})
C ({(1,1),(2,2),(3,3)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
B. ({1,2,3})
Step 1
Concept
The domain is formed by the first components (1,2,3). The identity relation has the whole set (A) as its domain.
Step 2
Why this answer is correct
The correct answer is B. ({1,2,3}). The domain is formed by the first components (1,2,3). The identity relation has the whole set (A) as its domain.
Step 3
Exam Tip
प्रांत पहले अवयवों (1,2,3) से बनता है। तत्समक संबंध का प्रांत पूरा (A) होता है।
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यदि \(A=\{1,2\}\) और \(B=\{2,3\}\), \(R=\{(a,b):a\le b\}\) है, तो (R) में कुल कितने युग्म हैं?
If \(A=\{1,2\}\), \(B=\{2,3\}\), and \(R=\{(a,b):a\le b\}\), how many pairs are in (R)?
#inequality relation
#counting
#relations
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
All possible pairs ((1,2),(1,3),(2,2),(2,3)) satisfy \(a\le b\). Therefore there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is C. (4). All possible pairs ((1,2),(1,3),(2,2),(2,3)) satisfy \(a\le b\). Therefore there are (4) pairs.
Step 3
Exam Tip
सभी संभावित युग्म ((1,2),(1,3),(2,2),(2,3)) शर्त \(a\le b\) पूरी करते हैं। इसलिए कुल (4) युग्म हैं।
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यदि \(R=\{(1,2),(2,1),(3,3)\}\) है, तो \(R^{-1}\) में कौन-सा युग्म अवश्य होगा?
If \(R=\{(1,2),(2,1),(3,3)\}\), which pair must be in \(R^{-1}\)?
#inverse relation
#relations
#ordered pairs
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A ((2,1))
B ((1,1))
C ((2,2))
D ((3,1))
Explanation opens after your attempt
Correct Answer
A. ((2,1))
Step 1
Concept
Since \((1,2)\in R\), we have \((2,1)\in R^{-1}\). Every pair is reversed in the inverse relation.
Step 2
Why this answer is correct
The correct answer is A. ((2,1)). Since \((1,2)\in R\), we have \((2,1)\in R^{-1}\). Every pair is reversed in the inverse relation.
Step 3
Exam Tip
क्योंकि \((1,2)\in R\), इसलिए \((2,1)\in R^{-1}\)। प्रतिलोम में हर युग्म उलट जाता है।
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यदि \(A=\{1,2,3,4\}\) और \(R=\{(a,b):b=a+1\}\) है, जहां \(a,b\in A\), तो कौन-सा युग्म (R) में है?
If \(A=\{1,2,3,4\}\) and \(R=\{(a,b):b=a+1\}\), where \(a,b\in A\), which pair is in (R)?
#set builder
#successor relation
#relations
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A ((1,1))
B ((2,3))
C ((3,2))
D ((4,5))
Explanation opens after your attempt
Correct Answer
B. ((2,3))
Step 1
Concept
In ((2,3)), (3=2+1), and both components are in (A). ((4,5)) is wrong because \(5\notin A\).
Step 2
Why this answer is correct
The correct answer is B. ((2,3)). In ((2,3)), (3=2+1), and both components are in (A). ((4,5)) is wrong because \(5\notin A\).
Step 3
Exam Tip
((2,3)) में (3=2+1) और दोनों अवयव (A) में हैं। ((4,5)) गलत है क्योंकि \(5\notin A\)।
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यदि \(R\subseteq A\times B\) है, तो (R) किससे किसमें संबंध कहलाता है?
If \(R\subseteq A\times B\), then (R) is called a relation from which set to which set?
#relation from A to B
#cartesian product
#basic
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A (B) से (A) में / From (B) to (A)
B (A) से (B) में / From (A) to (B)
C (A) से (A) में / From (A) to (A)
D (B) से (B) में / From (B) to (B)
Explanation opens after your attempt
Correct Answer
B. (A) से (B) में / From (A) to (B)
Step 1
Concept
When \(R\subseteq A\times B\), it is a relation from the first set (A) to the second set (B). Never change the order.
Step 2
Why this answer is correct
The correct answer is B. (A) से (B) में / From (A) to (B). When \(R\subseteq A\times B\), it is a relation from the first set (A) to the second set (B). Never change the order.
Step 3
Exam Tip
जब \(R\subseteq A\times B\) हो, तो पहले समुच्चय (A) से दूसरे समुच्चय (B) में संबंध होता है। क्रम को कभी न बदलें।
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यदि \(R=\{(1,4),(2,4),(3,5)\}\) है, तो (4) किन अवयवों का प्रतिबिंब है?
If \(R=\{(1,4),(2,4),(3,5)\}\), then (4) is the image of which elements?
#image
#relations
#ordered pairs
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A ({1,2})
B ({3})
C ({4,5})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
A. ({1,2})
Step 1
Concept
The second component (4) appears in ((1,4)) and ((2,4)). So (4) is the image of (1) and (2).
Step 2
Why this answer is correct
The correct answer is A. ({1,2}). The second component (4) appears in ((1,4)) and ((2,4)). So (4) is the image of (1) and (2).
Step 3
Exam Tip
दूसरा अवयव (4) युग्म ((1,4)) और ((2,4)) में आता है। इसलिए (4), (1) और (2) का प्रतिबिंब है।
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यदि \(A=\{1,2,3\}\) और \(R=\{(a,b):a\mid b\}\), जहां \(a,b\in A\), तो कौन-सा युग्म (R) में नहीं है?
If \(A=\{1,2,3\}\) and \(R=\{(a,b):a\mid b\}\), where \(a,b\in A\), which pair is not in (R)?
#divisibility relation
#relations
#checking condition
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A ((1,3))
B ((2,2))
C ((3,3))
D ((2,3))
Explanation opens after your attempt
Correct Answer
D. ((2,3))
Step 1
Concept
The pair ((2,3)) is wrong because (2) does not divide (3). In divisibility relations, the remainder must be zero.
Step 2
Why this answer is correct
The correct answer is D. ((2,3)). The pair ((2,3)) is wrong because (2) does not divide (3). In divisibility relations, the remainder must be zero.
Step 3
Exam Tip
((2,3)) गलत है क्योंकि (2) संख्या (3) को विभाजित नहीं करती। विभाज्यता संबंध में शेषफल शून्य होना चाहिए।
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यदि \(A=\{1,2,3\}\) पर \(R=A\times A\) है, तो (R) कौन-सा संबंध है?
If \(R=A\times A\) on \(A=\{1,2,3\}\), what type of relation is (R)?
#universal relation
#relations
#basic
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A खाली संबंध / Empty relation
B तत्समक संबंध / Identity relation
C सार्वत्रिक संबंध / Universal relation
D प्रतिलोम संबंध / Inverse relation
Explanation opens after your attempt
Correct Answer
C. सार्वत्रिक संबंध / Universal relation
Step 1
Concept
A relation containing all pairs of \(A\times A\) is the universal relation. It is the largest possible relation.
Step 2
Why this answer is correct
The correct answer is C. सार्वत्रिक संबंध / Universal relation. A relation containing all pairs of \(A\times A\) is the universal relation. It is the largest possible relation.
Step 3
Exam Tip
\(A\times A\) के सभी युग्मों को लेने वाला संबंध सार्वत्रिक संबंध होता है। यह सबसे बड़ा संभव संबंध है।
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यदि \(R=\varnothing\) है, तो उसका प्रांत क्या होगा?
If \(R=\varnothing\), what will be its domain?
#empty relation
#domain
#relations
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A \(\varnothing\)
B ({0})
C ({1})
D निर्धारित नहीं / Not defined
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
The empty relation has no ordered pair, so it has no first component. Hence the domain is \(\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). The empty relation has no ordered pair, so it has no first component. Hence the domain is \(\varnothing\).
Step 3
Exam Tip
खाली संबंध में कोई क्रमित युग्म नहीं है, इसलिए कोई पहला अवयव भी नहीं है। अतः प्रांत \(\varnothing\) है।
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यदि \(R=\{(1,2),(2,3),(3,1)\}\) है, तो (R) का प्रांत और परिसर क्रमशः क्या हैं?
If \(R=\{(1,2),(2,3),(3,1)\}\), what are the domain and range respectively?
#domain
#range
#relations
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A ({1,2,3},{1,2,3})
B ({1,2},{2,3})
C ({2,3,1},{1})
D ({(1,2),(2,3)},{3,1})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3},{1,2,3})
Step 1
Concept
The first components are (1,2,3), and the second components are (2,3,1), which as a set is ({1,2,3}). Be careful with set order.
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3},{1,2,3}). The first components are (1,2,3), and the second components are (2,3,1), which as a set is ({1,2,3}). Be careful with set order.
Step 3
Exam Tip
पहले अवयव (1,2,3) हैं और दूसरे अवयव (2,3,1) हैं, जो समुच्चय के रूप में ({1,2,3}) है। क्रम लिखने में ध्यान रखें।
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\(यदि (A={1,2,3,4}) और (R={(a,b):a+b\) is even\(}), जहां (a,b\in A), तो कौन-सा युग्म (R) में है\)?
\(If (A={1,2,3,4}) and (R={(a,b):a+b\) is even\(}), where (a,b\in A), which pair is in (R)\)?
#parity relation
#relations
#ordered pairs
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A ((1,2))
B ((2,3))
C ((3,4))
D ((2,4))
Explanation opens after your attempt
Correct Answer
D. ((2,4))
Step 1
Concept
In ((2,4)), (2+4=6), which is even. For an even sum, both components have the same parity.
Step 2
Why this answer is correct
The correct answer is D. ((2,4)). In ((2,4)), (2+4=6), which is even. For an even sum, both components have the same parity.
Step 3
Exam Tip
((2,4)) में (2+4=6), जो सम है। सम योग के लिए दोनों अवयव समान parity के होते हैं।
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यदि \(R=\{(1,1),(1,2),(2,2)\}\) है, तो (R) का परिसर क्या है?
If \(R=\{(1,1),(1,2),(2,2)\}\), what is the range of (R)?
#range
#relations
#duplicate values
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A ({1})
B ({2})
C ({1,2})
D ({(1,1),(2,2)})
Explanation opens after your attempt
Correct Answer
C. ({1,2})
Step 1
Concept
The second components are (1,2,2), and after removing repetition we get ({1,2}). The range uses only second components.
Step 2
Why this answer is correct
The correct answer is C. ({1,2}). The second components are (1,2,2), and after removing repetition we get ({1,2}). The range uses only second components.
Step 3
Exam Tip
दूसरे अवयव (1,2,2) हैं और दोहराव हटाने पर ({1,2}) मिलता है। परिसर में केवल दूसरे अवयव लिए जाते हैं।
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यदि \(A=\{a,b,c\}\) है, तो (A) पर तत्समक संबंध में कितने युग्म होंगे?
If \(A=\{a,b,c\}\), how many pairs will be in the identity relation on (A)?
#identity relation
#counting
#relations
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A (1)
B (2)
C (3)
D (9)
Explanation opens after your attempt
Step 1
Concept
In an identity relation, each element gives one pair ((x,x)). Therefore (3) elements give (3) pairs.
Step 2
Why this answer is correct
The correct answer is C. (3). In an identity relation, each element gives one pair ((x,x)). Therefore (3) elements give (3) pairs.
Step 3
Exam Tip
तत्समक संबंध में हर अवयव के लिए एक युग्म ((x,x)) होता है। इसलिए (3) अवयवों के लिए (3) युग्म होंगे।
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यदि \(A=\{1,2\}\) और \(R=\{(1,2),(2,1)\}\) है, तो \(R^{-1}\) कैसा है?
If \(A=\{1,2\}\) and \(R=\{(1,2),(2,1)\}\), what is \(R^{-1}\)?
#inverse relation
#symmetric pattern
#relations
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A \(R^{-1}=R\)
B \(R^{-1}=\varnothing\)
C \(R^{-1}=A\times A\)
D (R^{-1}={(1,1),(2,2)})
Explanation opens after your attempt
Correct Answer
A. \(R^{-1}=R\)
Step 1
Concept
On reversing both pairs, we get the same set again. Hence \(R^{-1}=R\).
Step 2
Why this answer is correct
The correct answer is A. \(R^{-1}=R\). On reversing both pairs, we get the same set again. Hence \(R^{-1}=R\).
Step 3
Exam Tip
दोनों युग्म उलटने पर फिर वही समुच्चय मिलता है। इसलिए \(R^{-1}=R\)।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,4,9\}\), \(R=\{(x,y):y=x^2\}\) है, तो कौन-सा युग्म (R) में नहीं है?
If \(A=\{1,2,3\}\), \(B=\{1,4,9\}\), and \(R=\{(x,y):y=x^2\}\), which pair is not in (R)?
#rule based relation
#squares
#relations
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A ((1,1))
B ((2,4))
C ((3,9))
D ((2,9))
Explanation opens after your attempt
Correct Answer
D. ((2,9))
Step 1
Concept
The pair ((2,9)) is wrong because \(2^2=4\), not (9). In rule-based relations, apply the given formula directly.
Step 2
Why this answer is correct
The correct answer is D. ((2,9)). The pair ((2,9)) is wrong because \(2^2=4\), not (9). In rule-based relations, apply the given formula directly.
Step 3
Exam Tip
((2,9)) गलत है क्योंकि \(2^2=4\), (9) नहीं। नियम आधारित संबंधों में दिए गए सूत्र को सीधे लगाएं।
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यदि \(R=\{(1,a),(2,b),(3,c)\}\) है, तो (R) के प्रतिलोम का प्रांत क्या होगा?
If \(R=\{(1,a),(2,b),(3,c)\}\), what will be the domain of \(R^{-1}\)?
#inverse relation
#domain
#range
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A ({1,2,3})
B ({a,b,c})
C ({(a,1),(b,2),(c,3)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
B. ({a,b,c})
Step 1
Concept
In the inverse relation, second components become first components. Hence the domain of \(R^{-1}\) is the range of (R), which is ({a,b,c}).
Step 2
Why this answer is correct
The correct answer is B. ({a,b,c}). In the inverse relation, second components become first components. Hence the domain of \(R^{-1}\) is the range of (R), which is ({a,b,c}).
Step 3
Exam Tip
प्रतिलोम में दूसरे अवयव पहले बन जाते हैं। इसलिए \(R^{-1}\) का प्रांत (R) के परिसर ({a,b,c}) के बराबर है।
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यदि \(A=\{1,2,3,4\}\) और \(R=\{(x,y):x=y-1\}\), जहां \(x,y\in A\), तो कौन-सा युग्म (R) में है?
If \(A=\{1,2,3,4\}\) and \(R=\{(x,y):x=y-1\}\), where \(x,y\in A\), which pair is in (R)?
#set builder
#relations
#linear rule
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A ((1,2))
B ((2,1))
C ((4,3))
D ((1,1))
Explanation opens after your attempt
Correct Answer
A. ((1,2))
Step 1
Concept
In ((1,2)), (1=2-1) is true. Substitute the components correctly for (x) and (y).
Step 2
Why this answer is correct
The correct answer is A. ((1,2)). In ((1,2)), (1=2-1) is true. Substitute the components correctly for (x) and (y).
Step 3
Exam Tip
((1,2)) में (1=2-1) सत्य है। शर्त में (x) और (y) की जगह सही अवयव रखें।
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यदि \(A=\{1,2\}\) है, तो (A) पर खाली संबंध और सार्वत्रिक संबंध में युग्मों की संख्या क्रमशः क्या है?
If \(A=\{1,2\}\), what are the numbers of pairs in the empty relation and universal relation on (A) respectively?
#empty relation
#universal relation
#counting
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A ((0,2))
B ((0,4))
C ((2,4))
D ((4,0))
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Correct Answer
B. ((0,4))
Step 1
Concept
The empty relation has (0) pairs, and the universal relation \(A\times A\) has \(2^2=4\) pairs. Remember these two extreme relations.
Step 2
Why this answer is correct
The correct answer is B. ((0,4)). The empty relation has (0) pairs, and the universal relation \(A\times A\) has \(2^2=4\) pairs. Remember these two extreme relations.
Step 3
Exam Tip
खाली संबंध में (0) युग्म होते हैं और सार्वत्रिक संबंध \(A\times A\) में \(2^2=4\) युग्म होते हैं। दोनों चरम संबंधों को याद रखें।
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यदि \(R=\{(1,2),(1,3),(2,3)\}\) है, तो (1) के कितने प्रतिबिंब हैं?
If \(R=\{(1,2),(1,3),(2,3)\}\), how many images does (1) have?
#image
#counting
#relations
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A (1)
B (2)
C (3)
D (0)
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Step 1
Concept
The element (1) appears as the first component in ((1,2)) and ((1,3)). So it has (2) images, (2) and (3).
Step 2
Why this answer is correct
The correct answer is B. (2). The element (1) appears as the first component in ((1,2)) and ((1,3)). So it has (2) images, (2) and (3).
Step 3
Exam Tip
(1) पहले अवयव के रूप में ((1,2)) और ((1,3)) में आता है। इसलिए उसके (2) प्रतिबिंब (2) और (3) हैं।
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यदि \(A=\{1,2,3\}\) और \(R=\{(a,b):a>b\}\), जहां \(a,b\in A\), तो (R) में कितने युग्म होंगे?
If \(A=\{1,2,3\}\) and \(R=\{(a,b):a>b\}\), where \(a,b\in A\), how many pairs will be in (R)?
#inequality relation
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A (1)
B (2)
C (3)
D (6)
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Step 1
Concept
The correct pairs are ((2,1),(3,1),(3,2)). Therefore there are (3) pairs in total.
Step 2
Why this answer is correct
The correct answer is C. (3). The correct pairs are ((2,1),(3,1),(3,2)). Therefore there are (3) pairs in total.
Step 3
Exam Tip
सही युग्म ((2,1),(3,1),(3,2)) हैं। इसलिए कुल (3) युग्म हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो (A) से (B) में सबसे छोटा संबंध कौन-सा है?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), which is the smallest relation from (A) to (B)?
#empty relation
#smallest relation
#relations
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A \(A\times B\)
B \(\varnothing\)
C ({(1,4)})
D ({(3,5)})
Explanation opens after your attempt
Correct Answer
B. \(\varnothing\)
Step 1
Concept
The smallest subset is \(\varnothing\), so the empty relation is the smallest relation. It is a subset of every \(A\times B\).
Step 2
Why this answer is correct
The correct answer is B. \(\varnothing\). The smallest subset is \(\varnothing\), so the empty relation is the smallest relation. It is a subset of every \(A\times B\).
Step 3
Exam Tip
सबसे छोटा उपसमुच्चय \(\varnothing\) होता है, इसलिए खाली संबंध सबसे छोटा संबंध है। यह हर \(A\times B\) का उपसमुच्चय है।
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यदि \(A=\{1,2\}\) और \(B=\{3,4\}\) हैं, तो (A) से (B) में सबसे बड़ा संबंध कौन-सा है?
If \(A=\{1,2\}\) and \(B=\{3,4\}\), which is the largest relation from (A) to (B)?
#largest relation
#cartesian product
#relations
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A \(\varnothing\)
B \(A\cap B\)
C \(A\times B\)
D \(B\times A\)
Explanation opens after your attempt
Correct Answer
C. \(A\times B\)
Step 1
Concept
The largest relation is the whole \(A\times B\). It contains all possible ordered pairs.
Step 2
Why this answer is correct
The correct answer is C. \(A\times B\). The largest relation is the whole \(A\times B\). It contains all possible ordered pairs.
Step 3
Exam Tip
सबसे बड़ा संबंध पूरा \(A\times B\) होता है। इसमें सभी संभावित क्रमित युग्म शामिल होते हैं।
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यदि \(R=\{(1,5),(2,6),(4,8)\}\) है, तो कौन-सा अवयव (R) के प्रांत में नहीं है?
If \(R=\{(1,5),(2,6),(4,8)\}\), which element is not in the domain of (R)?
#domain
#relations
#ordered pairs
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The domain is ({1,2,4}), so (3) is not in it. For domain, look only at the first components.
Step 2
Why this answer is correct
The correct answer is C. (3). The domain is ({1,2,4}), so (3) is not in it. For domain, look only at the first components.
Step 3
Exam Tip
प्रांत ({1,2,4}) है, इसलिए (3) इसमें नहीं है। प्रांत में केवल पहले अवयव देखे जाते हैं।
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यदि \(R=\{(1,2),(2,2),(3,2)\}\) है, तो कौन-सा कथन सही है?
If \(R=\{(1,2),(2,2),(3,2)\}\), which statement is correct?
#range
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A प्रांत ({2}) है / The domain is ({2})
B परिसर ({2}) है / The range is ({2})
C सहप्रांत ({1,2,3}) है / The codomain is ({1,2,3})
D प्रांत \(\varnothing\) है / The domain is \(\varnothing\)
Explanation opens after your attempt
Correct Answer
B. परिसर ({2}) है / The range is ({2})
Step 1
Concept
All second components are (2), so the range is ({2}). The domain would be ({1,2,3}).
Step 2
Why this answer is correct
The correct answer is B. परिसर ({2}) है / The range is ({2}). All second components are (2), so the range is ({2}). The domain would be ({1,2,3}).
Step 3
Exam Tip
दूसरे अवयव सभी (2) हैं, इसलिए परिसर ({2}) है। प्रांत ({1,2,3}) होगा।
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यदि \(A=\{1,2,3\}\) और \(R=\{(x,y):x-y=0\}\), जहां \(x,y\in A\), तो (R) किसके बराबर है?
If \(A=\{1,2,3\}\) and \(R=\{(x,y):x-y=0\}\), where \(x,y\in A\), then (R) is equal to what?
#identity relation
#equation relation
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A ({(1,2),(2,3)})
B ({(1,1),(2,2),(3,3)})
C ({(1,3),(3,1)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
B. ({(1,1),(2,2),(3,3)})
Step 1
Concept
The equation (x-y=0) means (x=y). Therefore this is the identity relation on (A).
Step 2
Why this answer is correct
The correct answer is B. ({(1,1),(2,2),(3,3)}). The equation (x-y=0) means (x=y). Therefore this is the identity relation on (A).
Step 3
Exam Tip
(x-y=0) का अर्थ (x=y) है। इसलिए यह (A) पर तत्समक संबंध है।
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यदि \(A=\{1,2,3,4,5\}\) और \(R=\{(x,y):y=2x\}\), जहां \(x,y\in A\), तो (R) का परिसर क्या है?
If \(A=\{1,2,3,4,5\}\) and \(R=\{(x,y):y=2x\}\), where \(x,y\in A\), what is the range of (R)?
#range
#rule based relation
#relations
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A ({2,4})
B ({1,2})
C ({2,4,6,8,10})
D ({1,2,3,4,5})
Explanation opens after your attempt
Correct Answer
A. ({2,4})
Step 1
Concept
For (x=1), (y=2), and for (x=2), (y=4); after that \(y\notin A\). So the range is ({2,4}).
Step 2
Why this answer is correct
The correct answer is A. ({2,4}). For (x=1), (y=2), and for (x=2), (y=4); after that \(y\notin A\). So the range is ({2,4}).
Step 3
Exam Tip
(x=1) पर (y=2) और (x=2) पर (y=4) मिलता है; आगे \(y\notin A\) हो जाता है। इसलिए परिसर ({2,4}) है।
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यदि \(R=\{(a,1),(b,2),(c,3)\}\) है, तो \(R^{-1}\) में ((2,b)) के बारे में क्या सही है?
If \(R=\{(a,1),(b,2),(c,3)\}\), what is true about ((2,b)) in \(R^{-1}\)?
#inverse relation
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A \((2,b)\in R^{-1}\)
B \((2,b)\notin R^{-1}\)
C \((b,2)\in R^{-1}\)
D \(R^{-1}=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \((2,b)\in R^{-1}\)
Step 1
Concept
Since \((b,2)\in R\), reversing it gives \((2,b)\in R^{-1}\). Changing the order is the key point in inverse relations.
Step 2
Why this answer is correct
The correct answer is A. \((2,b)\in R^{-1}\). Since \((b,2)\in R\), reversing it gives \((2,b)\in R^{-1}\). Changing the order is the key point in inverse relations.
Step 3
Exam Tip
क्योंकि \((b,2)\in R\), इसलिए उलटकर \((2,b)\in R^{-1}\) होगा। प्रतिलोम में क्रम बदलना मुख्य बात है।
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यदि \(A=\{1,2,3\}\) और \(R=\{(1,2),(2,3)\}\) है, तो कौन-सा अवयव (R) के प्रांत में है लेकिन परिसर में नहीं है?
If \(A=\{1,2,3\}\) and \(R=\{(1,2),(2,3)\}\), which element is in the domain of (R) but not in the range?
#domain
#range
#set difference
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A (1)
B (2)
C (3)
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The domain is ({1,2}), and the range is ({2,3}). Only (1) is in the domain but not in the range.
Step 2
Why this answer is correct
The correct answer is A. (1). The domain is ({1,2}), and the range is ({2,3}). Only (1) is in the domain but not in the range.
Step 3
Exam Tip
प्रांत ({1,2}) और परिसर ({2,3}) है। इनमें केवल (1) प्रांत में है लेकिन परिसर में नहीं है।
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यदि \(A=\{1,2,3\}\) और \(R=\{(a,b):a+b=4\}\), जहां \(a,b\in A\), तो (R) का सही रूप क्या है?
If \(A=\{1,2,3\}\) and \(R=\{(a,b):a+b=4\}\), where \(a,b\in A\), what is the correct form of (R)?
#set builder
#sum relation
#relations
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A ({(1,3),(2,2),(3,1)})
B ({(1,2),(2,1)})
C ({(1,1),(2,2),(3,3)})
D ({(3,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(2,2),(3,1)})
Step 1
Concept
The pairs whose sum is (4) are ((1,3),(2,2),(3,1)). In such questions, check all possible pairs systematically.
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,2),(3,1)}). The pairs whose sum is (4) are ((1,3),(2,2),(3,1)). In such questions, check all possible pairs systematically.
Step 3
Exam Tip
जिन युग्मों का योग (4) है वे ((1,3),(2,2),(3,1)) हैं। ऐसे प्रश्नों में सभी संभावित युग्म व्यवस्थित रूप से जांचें।
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