Since (n\(A\times B\)=3\times 2=6), the number of relations is \(2^6=64\). In exams, first count the elements of \(A\times B\).
Step 2
Why this answer is correct
The correct answer is C. (64). Since (n\(A\times B\)=3\times 2=6), the number of relations is \(2^6=64\). In exams, first count the elements of \(A\times B\).
Step 3
Exam Tip
क्योंकि (n\(A\times B\)=3\times 2=6) और संबंधों की संख्या \(2^6=64\) होती है। परीक्षा में पहले \(A\times B\) के अवयव गिनें।
A relation on (A) is a subset of \(A\times A\), and (n\(A\times A\)=16). Therefore total relations are \(2^{16}\).
Step 2
Why this answer is correct
The correct answer is C. \(2^{16}\). A relation on (A) is a subset of \(A\times A\), and (n\(A\times A\)=16). Therefore total relations are \(2^{16}\).
Step 3
Exam Tip
(A) पर संबंध \(A\times A\) का उपसमुच्चय होता है और (n\(A\times A\)=16) है। इसलिए कुल संबंध \(2^{16}\) होंगे।
For every \(a\in A\), \((a,a)\in R\), so the relation is reflexive. Symmetry would also require ((2,1)).
Step 2
Why this answer is correct
The correct answer is A. प्रतिवर्ती / Reflexive. For every \(a\in A\), \((a,a)\in R\), so the relation is reflexive. Symmetry would also require ((2,1)).
Step 3
Exam Tip
हर \(a\in A\) के लिए \((a,a)\in R\) है इसलिए संबंध प्रतिवर्ती है। सममितता के लिए ((2,1)) भी चाहिए था।
For every \((a,b)\in R\), \((b,a)\in R\) is also present, so it is symmetric. To be reflexive, it needs ((1,1),(2,2),(3,3)).
Step 2
Why this answer is correct
The correct answer is B. सममित / Symmetric. For every \((a,b)\in R\), \((b,a)\in R\) is also present, so it is symmetric. To be reflexive, it needs ((1,1),(2,2),(3,3)).
Step 3
Exam Tip
हर \((a,b)\in R\) के साथ \((b,a)\in R\) भी है इसलिए यह सममित है। प्रतिवर्ती होने के लिए ((1,1),(2,2),(3,3)) चाहिए।
Since ((1,2)) and ((2,3)) imply ((1,3)), the relation shows transitivity. In exams, check matching middle elements.
Step 2
Why this answer is correct
The correct answer is C. सकर्मक / Transitive. Since ((1,2)) and ((2,3)) imply ((1,3)), the relation shows transitivity. In exams, check matching middle elements.
Step 3
Exam Tip
क्योंकि ((1,2)) और ((2,3)) से ((1,3)) भी संबंध में है इसलिए यह सकर्मक स्थिति दिखाता है। परीक्षा में मध्य अवयव मिलाकर जांचें।
Only pairs of the form ((a,a)) are present, so it is the identity relation. It is also reflexive and symmetric.
Step 2
Why this answer is correct
The correct answer is C. सर्वसम संबंध / Identity relation. Only pairs of the form ((a,a)) are present, so it is the identity relation. It is also reflexive and symmetric.
Step 3
Exam Tip
केवल ((a,a)) रूप के युग्म हैं इसलिए यह सर्वसम संबंध है। यह प्रतिवर्ती और सममित दोनों भी है।
In the inverse relation, the components of each ordered pair are interchanged. Thus ((1,3)) becomes ((3,1)).
Step 2
Why this answer is correct
The correct answer is A. \({(3,1),(5,2),(9,4)}\). In the inverse relation, the components of each ordered pair are interchanged. Thus ((1,3)) becomes ((3,1)).
Step 3
Exam Tip
विलोम संबंध में हर क्रमित युग्म के अवयवों का स्थान बदल जाता है। इसलिए ((1,3)) से ((3,1)) बनेगा।
In ((4,5)), \(5\notin A\), so it is not a pair of \(A\times A\). While forming a relation, both components must belong to the set.
Step 2
Why this answer is correct
The correct answer is D. \((4,5)\). In ((4,5)), \(5\notin A\), so it is not a pair of \(A\times A\). While forming a relation, both components must belong to the set.
Step 3
Exam Tip
((4,5)) में \(5\notin A\) है इसलिए यह \(A\times A\) का युग्म नहीं है। संबंध बनाते समय दोनों अवयव समुच्चय में होने चाहिए।
A. \(A\times B\) का कोई भी उपसमुच्चय/Any subset of \(A\times B\)
Step 1
Concept
A relation from (A) to (B) is any subset of \(A\times B\). This basic definition is often asked directly.
Step 2
Why this answer is correct
The correct answer is A. \(A\times B\) का कोई भी उपसमुच्चय / Any subset of \(A\times B\). A relation from (A) to (B) is any subset of \(A\times B\). This basic definition is often asked directly.
Step 3
Exam Tip
(A) से (B) तक संबंध \(A\times B\) का कोई भी उपसमुच्चय होता है। यह मूल परिभाषा अक्सर सीधे पूछी जाती है।
Total relations are \(2^{5\times 3}=2^{15}\), and removing the empty relation gives \(2^{15}-1\). For non-empty relations, remember to subtract (1).
Step 2
Why this answer is correct
The correct answer is B. \(2^{15}-1\). Total relations are \(2^{5\times 3}=2^{15}\), and removing the empty relation gives \(2^{15}-1\). For non-empty relations, remember to subtract (1).
Step 3
Exam Tip
कुल संबंध \(2^{5\times 3}=2^{15}\) हैं और रिक्त संबंध हटाने पर \(2^{15}-1\) मिलते हैं। गैर रिक्त के लिए (1) घटाना न भूलें।
There is no ordered pair in \(\varnothing\), so it is the empty relation. Since \(A\neq\varnothing\), it is not reflexive.
Step 2
Why this answer is correct
The correct answer is B. (R) रिक्त संबंध है / (R) is the empty relation. There is no ordered pair in \(\varnothing\), so it is the empty relation. Since \(A\neq\varnothing\), it is not reflexive.
Step 3
Exam Tip
\(\varnothing\) में कोई क्रमित युग्म नहीं होता इसलिए यह रिक्त संबंध है। \(A\neq\varnothing\) होने पर यह प्रतिवर्ती नहीं है।
When a relation contains all pairs of \(A\times A\), it is the universal relation. It contains every possible ordered pair.
Step 2
Why this answer is correct
The correct answer is B. सार्वभौम संबंध / Universal relation. When a relation contains all pairs of \(A\times A\), it is the universal relation. It contains every possible ordered pair.
Step 3
Exam Tip
जब संबंध \(A\times A\) के सभी युग्मों को लेता है तो वह सार्वभौम संबंध होता है। इसमें हर संभव क्रमित युग्म होता है।
B. क्योंकि \((3,3)\notin R\)/Because \((3,3)\notin R\)
Step 1
Concept
For reflexivity, \((a,a)\in R\) must hold for every \(a\in A\). Here ((3,3)) is missing.
Step 2
Why this answer is correct
The correct answer is B. क्योंकि \((3,3)\notin R\) / Because \((3,3)\notin R\). For reflexivity, \((a,a)\in R\) must hold for every \(a\in A\). Here ((3,3)) is missing.
Step 3
Exam Tip
प्रतिवर्ती होने के लिए हर \(a\in A\) पर \((a,a)\in R\) होना चाहिए। यहां ((3,3)) अनुपस्थित है।
A. हर युग्म का उल्टा युग्म भी है/The reverse of every pair is also present
Step 1
Concept
The pair ((2,1)) is present with ((1,2)), and ((3,3)) is its own reverse. Therefore the relation is symmetric.
Step 2
Why this answer is correct
The correct answer is A. हर युग्म का उल्टा युग्म भी है / The reverse of every pair is also present. The pair ((2,1)) is present with ((1,2)), and ((3,3)) is its own reverse. Therefore the relation is symmetric.
Step 3
Exam Tip
((1,2)) के साथ ((2,1)) है और ((3,3)) अपना उल्टा स्वयं है। इसलिए संबंध सममित है।
It is reflexive because ((1,1),(2,2),(3,3)) are present, and symmetric because the reverse of ((1,2)) is ((2,1)). Hence it satisfies both.
Step 2
Why this answer is correct
The correct answer is A. प्रतिवर्ती और सममित / Reflexive and symmetric. It is reflexive because ((1,1),(2,2),(3,3)) are present, and symmetric because the reverse of ((1,2)) is ((2,1)). Hence it satisfies both.
Step 3
Exam Tip
((1,1),(2,2),(3,3)) हैं इसलिए यह प्रतिवर्ती है और ((1,2)) का उल्टा ((2,1)) भी है। इसलिए यह सममित भी है।
A. क्योंकि \((1,3)\notin R\)/Because \((1,3)\notin R\)
Step 1
Concept
Since ((1,2)) and ((2,3)) are present, transitivity requires ((1,3)). It is missing, so transitivity fails.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \((1,3)\notin R\) / Because \((1,3)\notin R\). Since ((1,2)) and ((2,3)) are present, transitivity requires ((1,3)). It is missing, so transitivity fails.
Step 3
Exam Tip
((1,2)) और ((2,3)) मौजूद हैं तो सकर्मकता के लिए ((1,3)) चाहिए। यह अनुपस्थित है इसलिए सकर्मकता असफल है।
A. हाँ क्योंकि यह प्रतिवर्ती सममित और सकर्मक है/Yes because it is reflexive symmetric and transitive
Step 1
Concept
It is reflexive and the reverses of ((1,2),(2,1)) are present. The required transitive pairs ((1,1),(2,2)) are also present.
Step 2
Why this answer is correct
The correct answer is A. हाँ क्योंकि यह प्रतिवर्ती सममित और सकर्मक है / Yes because it is reflexive symmetric and transitive. It is reflexive and the reverses of ((1,2),(2,1)) are present. The required transitive pairs ((1,1),(2,2)) are also present.
Step 3
Exam Tip
यह प्रतिवर्ती है और ((1,2),(2,1)) के उल्टे मौजूद हैं। साथ ही आवश्यक सकर्मक युग्म ((1,1),(2,2)) मौजूद हैं।
The key feature of a reflexive relation is that every element is related to itself. Remember this definition directly.
Step 2
Why this answer is correct
The correct answer is A. प्रतिवर्ती संबंध / Reflexive relation. The key feature of a reflexive relation is that every element is related to itself. Remember this definition directly.
Step 3
Exam Tip
प्रतिवर्ती संबंध की यही पहचान है कि हर अवयव स्वयं से संबंधित हो। इस परिभाषा को सीधे याद रखें।
A. यदि \((a,b)\in R\Rightarrow (b,a)\in R\)/If \((a,b)\in R\Rightarrow (b,a)\in R\)
Step 1
Concept
In symmetry, the reverse of every pair must also be in the relation. This is the correct condition among the options.
Step 2
Why this answer is correct
The correct answer is A. यदि \((a,b)\in R\Rightarrow (b,a)\in R\) / If \((a,b)\in R\Rightarrow (b,a)\in R\). In symmetry, the reverse of every pair must also be in the relation. This is the correct condition among the options.
Step 3
Exam Tip
सममितता में हर युग्म का उल्टा युग्म भी संबंध में होना चाहिए। विकल्पों में यही सही शर्त है।
A. यदि \((a,b)\in R\) और \((b,c)\in R\Rightarrow (a,c)\in R\)/If \((a,b)\in R\) and \((b,c)\in R\Rightarrow (a,c)\in R\)
Step 1
Concept
In transitivity, two connected pairs must imply a pair between the first and last elements. Therefore check the presence of ((a,c)).
Step 2
Why this answer is correct
The correct answer is A. यदि \((a,b)\in R\) और \((b,c)\in R\Rightarrow (a,c)\in R\) / If \((a,b)\in R\) and \((b,c)\in R\Rightarrow (a,c)\in R\). In transitivity, two connected pairs must imply a pair between the first and last elements. Therefore check the presence of ((a,c)).
Step 3
Exam Tip
सकर्मकता में दो जुड़े युग्मों से पहला और अंतिम अवयव जुड़ना चाहिए। इसलिए ((a,c)) की उपस्थिति जांचें।
Every \(x\leq x\), so it is reflexive, and \(x\leq y\leq z\) gives \(x\leq z\). It is generally not symmetric.
Step 2
Why this answer is correct
The correct answer is A. प्रतिवर्ती और सकर्मक / Reflexive and transitive. Every \(x\leq x\), so it is reflexive, and \(x\leq y\leq z\) gives \(x\leq z\). It is generally not symmetric.
Step 3
Exam Tip
हर \(x\leq x\) है इसलिए प्रतिवर्ती है और \(x\leq y\leq z\) से \(x\leq z\) मिलता है। यह सामान्यतः सममित नहीं है।
All elements of the set are even, so the difference of any two elements is divisible by (2). Hence \(R=A\times A\).
Step 2
Why this answer is correct
The correct answer is A. सार्वभौम संबंध / Universal relation. All elements of the set are even, so the difference of any two elements is divisible by (2). Hence \(R=A\times A\).
Step 3
Exam Tip
समुच्चय के सभी अवयव सम हैं इसलिए किसी भी दो अवयवों का अंतर (2) से विभाज्य है। अतः \(R=A\times A\) है।
A relation from (A) to (B) is always a subset of \(A\times B\). Order matters, so \(B\times A\) can be different.
Step 2
Why this answer is correct
The correct answer is A. \(A\times B\). A relation from (A) to (B) is always a subset of \(A\times B\). Order matters, so \(B\times A\) can be different.
Step 3
Exam Tip
(A) से (B) तक संबंध सदैव \(A\times B\) का उपसमुच्चय होता है। क्रम महत्वपूर्ण है इसलिए \(B\times A\) अलग हो सकता है।
The first components give the domain ({1,2}), and the second components give the range ({2,3}). Write sets after removing repetitions.
Step 2
Why this answer is correct
The correct answer is A. \({1,2}) और ({2,3}) / ({1,2}) and ({2,3}). The first components give the domain ({1,2}), and the second components give the range ({2,3}). Write sets after removing repetitions.
Step 3
Exam Tip
पहले अवयवों से प्रांत ({1,2}) और दूसरे अवयवों से परिसर ({2,3}) है। दोहराव हटाकर समुच्चय लिखें।
A. क्योंकि यह सममित नहीं है/Because it is not symmetric
Step 1
Concept
Here \((1,2)\in R\) but \((2,1)\notin R\), so it is not symmetric. Equivalence needs reflexive, symmetric, and transitive properties.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि यह सममित नहीं है / Because it is not symmetric. Here \((1,2)\in R\) but \((2,1)\notin R\), so it is not symmetric. Equivalence needs reflexive, symmetric, and transitive properties.
Step 3
Exam Tip
\((1,2)\in R\) है लेकिन \((2,1)\notin R\) है इसलिए यह सममित नहीं है। तुल्यता के लिए प्रतिवर्ती सममित और सकर्मक तीनों चाहिए।
The odd elements in (A) are (1) and (3), so all ordered pairs are formed from them. Check both components in the condition.
Step 2
Why this answer is correct
The correct answer is A. \({(1,1),(1,3),(3,1),(3,3)}\). The odd elements in (A) are (1) and (3), so all ordered pairs are formed from them. Check both components in the condition.
Step 3
Exam Tip
(A) में विषम अवयव (1) और (3) हैं इसलिए इन्हीं से सभी क्रमित युग्म बनेंगे। शर्त में दोनों अवयवों को जांचें।
