Since \(|A\times B|=3\times2=6\), the number of relations is \(2^6\). In exams, first find the number of elements in \(A\times B\).
Step 2
Why this answer is correct
The correct answer is B. \(2^6\). Since \(|A\times B|=3\times2=6\), the number of relations is \(2^6\). In exams, first find the number of elements in \(A\times B\).
Step 3
Exam Tip
क्योंकि \(|A\times B|=3\times2=6\), इसलिए संबंधों की संख्या \(2^6\) होगी। परीक्षा में पहले \(A\times B\) के अवयवों की संख्या निकालें।
The domain is the set of first components of the ordered pairs, so ( {1,2,3} ) is correct. In exams, look at the first entries.
Step 2
Why this answer is correct
The correct answer is B. ( {1,2,3} ). The domain is the set of first components of the ordered pairs, so ( {1,2,3} ) is correct. In exams, look at the first entries.
Step 3
Exam Tip
प्रांत ordered pairs के पहले घटकों का समुच्चय होता है, इसलिए ( {1,2,3} ) सही है। परीक्षा में पहले स्थान वाले अवयव देखें।
The possible pairs are ( (1,2),(1,3),(1,4),(2,3),(2,4),(3,4) ), so the count is (6). In exams, list pairs systematically.
Step 2
Why this answer is correct
The correct answer is C. (6). The possible pairs are ( (1,2),(1,3),(1,4),(2,3),(2,4),(3,4) ), so the count is (6). In exams, list pairs systematically.
Step 3
Exam Tip
संभव युग्म ( (1,2),(1,3),(1,4),(2,3),(2,4),(3,4) ) हैं, इसलिए संख्या (6) है। परीक्षा में क्रमबद्ध तरीके से युग्म लिखें।
Each element is related only to itself, so it is the identity relation. In exams, identify pairs of the form ( (a,a) ).
Step 2
Why this answer is correct
The correct answer is B. सर्वसम संबंध / Identity relation. Each element is related only to itself, so it is the identity relation. In exams, identify pairs of the form ( (a,a) ).
Step 3
Exam Tip
हर अवयव केवल अपने आप से संबंधित है, इसलिए यह सर्वसम संबंध है। परीक्षा में ( (a,a) ) वाले युग्म पहचानें।
A universal relation contains all pairs of \(A\times A\), and \(|A\times A|=2\times2=4\). In exams, treat universal relation as the complete Cartesian product.
Step 2
Why this answer is correct
The correct answer is C. (4). A universal relation contains all pairs of \(A\times A\), and \(|A\times A|=2\times2=4\). In exams, treat universal relation as the complete Cartesian product.
Step 3
Exam Tip
सार्वत्रिक संबंध \(A\times A\) के सभी युग्म रखता है और \(|A\times A|=2\times2=4\)। परीक्षा में universal relation को पूरा Cartesian product मानें।
In the inverse relation, the components of each ordered pair are interchanged. Hence (R^{-1}={(3,1),(4,2),(5,3)}).
Step 2
Why this answer is correct
The correct answer is A. ( {(3,1),(4,2),(5,3)} ). In the inverse relation, the components of each ordered pair are interchanged. Hence (R^{-1}={(3,1),(4,2),(5,3)}).
Step 3
Exam Tip
व्युत्क्रम संबंध में हर ordered pair के घटक बदल जाते हैं। इसलिए (R^{-1}={(3,1),(4,2),(5,3)})।
B. क्योंकि \((3,3)\notin R\)/Because \((3,3)\notin R\)
Step 1
Concept
For a reflexive relation, \((a,a)\in R\) is required for every \(a\in A\). Here \((3,3)\notin R\), so it is not reflexive.
Step 2
Why this answer is correct
The correct answer is B. क्योंकि \((3,3)\notin R\) / Because \((3,3)\notin R\). For a reflexive relation, \((a,a)\in R\) is required for every \(a\in A\). Here \((3,3)\notin R\), so it is not reflexive.
Step 3
Exam Tip
परावर्ती संबंध के लिए हर \(a\in A\) पर \((a,a)\in R\) चाहिए। यहाँ \((3,3)\notin R\), इसलिए यह परावर्ती नहीं है।
For every \((a,b)\in R\), \((b,a)\in R\) is also present, so (R) is symmetric. In exams, match each pair with its reverse.
Step 2
Why this answer is correct
The correct answer is A. यह सममित है / It is symmetric. For every \((a,b)\in R\), \((b,a)\in R\) is also present, so (R) is symmetric. In exams, match each pair with its reverse.
Step 3
Exam Tip
हर \((a,b)\in R\) के साथ \((b,a)\in R\) भी है, इसलिए (R) सममित है। परीक्षा में उल्टे ordered pair को मिलाएं।
C. दिए गए युग्मों के लिए यह संक्रामी शर्त पूरी करता है/It satisfies the transitive condition for the given pairs
Step 1
Concept
Since \((1,2)\in R\) and \((2,3)\in R\) imply \((1,3)\in R\). For transitivity, connect the middle element carefully.
Step 2
Why this answer is correct
The correct answer is C. दिए गए युग्मों के लिए यह संक्रामी शर्त पूरी करता है / It satisfies the transitive condition for the given pairs. Since \((1,2)\in R\) and \((2,3)\in R\) imply \((1,3)\in R\). For transitivity, connect the middle element carefully.
Step 3
Exam Tip
क्योंकि \((1,2)\in R\) और \((2,3)\in R\) होने पर \((1,3)\in R\) भी है। संक्रामकता में बीच वाले अवयव को ध्यान से जोड़ें।
If (a-b) is even, then (a) and (b) have the same parity. In exams, make odd and even groups separately.
Step 2
Why this answer is correct
The correct answer is A. समान parity / Same parity. If (a-b) is even, then (a) and (b) have the same parity. In exams, make odd and even groups separately.
Step 3
Exam Tip
यदि (a-b) सम है, तो (a) और (b) दोनों समान parity के होते हैं। परीक्षा में odd और even समूह अलग बनाएं।
A relation from (A) to (B) must be a subset of \(A\times B\). So first components must come from (A), and second components must come from (B).
Step 2
Why this answer is correct
The correct answer is A. ( {(1,4),(3,5)} ). A relation from (A) to (B) must be a subset of \(A\times B\). So first components must come from (A), and second components must come from (B).
Step 3
Exam Tip
(A) से (B) तक संबंध \(A\times B\) का उपसमुच्चय होना चाहिए। इसलिए पहले घटक (A) से और दूसरे घटक (B) से होने चाहिए।
Total relations are \(2^{|A\times B|}=2^{12}\), and removing the empty relation gives \(2^{12}-1\). In exams, remember to subtract (1) for non-empty relations.
Step 2
Why this answer is correct
The correct answer is B. \(2^{12}-1\). Total relations are \(2^{|A\times B|}=2^{12}\), and removing the empty relation gives \(2^{12}-1\). In exams, remember to subtract (1) for non-empty relations.
Step 3
Exam Tip
कुल संबंध \(2^{|A\times B|}=2^{12}\) हैं और रिक्त संबंध हटाने पर \(2^{12}-1\) मिलते हैं। परीक्षा में non-empty के लिए (1) घटाना न भूलें।
An empty relation has no ordered pair, so it is \( \varnothing \). In exams, keep empty relation and identity relation separate.
Step 2
Why this answer is correct
The correct answer is A. \( \varnothing \). An empty relation has no ordered pair, so it is \( \varnothing \). In exams, keep empty relation and identity relation separate.
Step 3
Exam Tip
रिक्त संबंध में कोई ordered pair नहीं होता, इसलिए यह \( \varnothing \) होता है। परीक्षा में empty relation और identity relation को अलग रखें।
According to (a=b+1), in ( (2,1) ), (2=1+1). In exams, do not interchange the roles of first and second components.
Step 2
Why this answer is correct
The correct answer is A. ( (2,1) ). According to (a=b+1), in ( (2,1) ), (2=1+1). In exams, do not interchange the roles of first and second components.
Step 3
Exam Tip
(a=b+1) के अनुसार ( (2,1) ) में (2=1+1) है। परीक्षा में पहले और दूसरे घटक की भूमिका न बदलें।
A. हाँ, क्योंकि \((1,1),(2,2),(3,3)\in R\)/Yes, because \((1,1),(2,2),(3,3)\in R\)
Step 1
Concept
For reflexivity, all pairs ( (a,a) ) are required, and they are present here. The extra pair ( (1,2) ) does not break reflexivity.
Step 2
Why this answer is correct
The correct answer is A. हाँ, क्योंकि \((1,1),(2,2),(3,3)\in R\) / Yes, because \((1,1),(2,2),(3,3)\in R\). For reflexivity, all pairs ( (a,a) ) are required, and they are present here. The extra pair ( (1,2) ) does not break reflexivity.
Step 3
Exam Tip
परावर्ती होने के लिए सभी ( (a,a) ) युग्म चाहिए और यहाँ वे मौजूद हैं। अतिरिक्त ( (1,2) ) होने से परावर्तिता खराब नहीं होती।
For symmetry, the reverse of ( (1,2) ) is ( (2,1) ), and the reverse of ( (2,2) ) is itself. ( (1,1) ) is not necessary.
Step 2
Why this answer is correct
The correct answer is D. ( (1,1) ). For symmetry, the reverse of ( (1,2) ) is ( (2,1) ), and the reverse of ( (2,2) ) is itself. ( (1,1) ) is not necessary.
Step 3
Exam Tip
सममिति के लिए ( (1,2) ) का उल्टा ( (2,1) ) और ( (2,2) ) का उल्टा वही ( (2,2) ) चाहिए। ( (1,1) ) जरूरी नहीं है।
The first component must be greater than some smaller (b), so (2,3,4) appear. (1) is not greater than any \(b\in A\).
Step 2
Why this answer is correct
The correct answer is B. ( {2,3,4} ). The first component must be greater than some smaller (b), so (2,3,4) appear. (1) is not greater than any \(b\in A\).
Step 3
Exam Tip
पहला घटक ऐसा होना चाहिए जो किसी छोटे (b) से बड़ा हो, इसलिए (2,3,4) आते हैं। (1) किसी भी \(b\in A\) से बड़ा नहीं है।
The second component must have some greater \(a\in A\), so (1,2,3) appear. There is no element in (A) greater than (4).
Step 2
Why this answer is correct
The correct answer is A. ( {1,2,3} ). The second component must have some greater \(a\in A\), so (1,2,3) appear. There is no element in (A) greater than (4).
Step 3
Exam Tip
दूसरा घटक ऐसा होना चाहिए जिससे बड़ा कोई \(a\in A\) मिले, इसलिए (1,2,3) आते हैं। (4) से बड़ा कोई अवयव (A) में नहीं है।
A. जब \(R\subseteq A\times B\)/When \(R\subseteq A\times B\)
Step 1
Concept
A relation from (A) to (B) is any subset of \(A\times B\). In exams, treat a relation as a set of ordered pairs.
Step 2
Why this answer is correct
The correct answer is A. जब \(R\subseteq A\times B\) / When \(R\subseteq A\times B\). A relation from (A) to (B) is any subset of \(A\times B\). In exams, treat a relation as a set of ordered pairs.
Step 3
Exam Tip
(A) से (B) तक संबंध \(A\times B\) का कोई भी उपसमुच्चय होता है। परीक्षा में संबंध को ordered pairs का समुच्चय मानें।
B. परावर्ती और सममित संबंध/Reflexive and symmetric relation
Step 1
Concept
It is reflexive because ( (1,1),(2,2),(3,3) ) are present, and symmetric because ( (1,2),(2,1) ) are paired. It is not universal because all pairs are not present.
Step 2
Why this answer is correct
The correct answer is B. परावर्ती और सममित संबंध / Reflexive and symmetric relation. It is reflexive because ( (1,1),(2,2),(3,3) ) are present, and symmetric because ( (1,2),(2,1) ) are paired. It is not universal because all pairs are not present.
Step 3
Exam Tip
( (1,1),(2,2),(3,3) ) होने से यह परावर्ती है और ( (1,2),(2,1) ) जोड़ी होने से सममित है। सभी युग्म न होने के कारण यह सार्वत्रिक नहीं है।
The pairs are ( (1,2),(2,1),(2,3),(3,2),(3,4),(4,3) ), so the count is (6). In exams, take both directions because of absolute difference.
Step 2
Why this answer is correct
The correct answer is C. (6). The pairs are ( (1,2),(2,1),(2,3),(3,2),(3,4),(4,3) ), so the count is (6). In exams, take both directions because of absolute difference.
Step 3
Exam Tip
युग्म ( (1,2),(2,1),(2,3),(3,2),(3,4),(4,3) ) हैं, इसलिए संख्या (6) है। परीक्षा में absolute difference के कारण दोनों दिशाएं लें।
(4) is a multiple of (2), so \((4,2)\in R\). In exams, understand the direction of multiple of and divides separately.
Step 2
Why this answer is correct
The correct answer is B. ( (4,2) ). (4) is a multiple of (2), so \((4,2)\in R\). In exams, understand the direction of multiple of and divides separately.
Step 3
Exam Tip
(4), (2) का multiple है, इसलिए \((4,2)\in R\)। परीक्षा में multiple of और divides की दिशा अलग समझें।
A. प्रांत ( {1,2,3} ), परिसर ( {4,5,6} )/Domain ( {1,2,3} ), range ( {4,5,6} )
Step 1
Concept
The domain is formed from first components and the range from second components. Therefore domain is ( {1,2,3} ) and range is ( {4,5,6} ).
Step 2
Why this answer is correct
The correct answer is A. प्रांत ( {1,2,3} ), परिसर ( {4,5,6} ) / Domain ( {1,2,3} ), range ( {4,5,6} ). The domain is formed from first components and the range from second components. Therefore domain is ( {1,2,3} ) and range is ( {4,5,6} ).
Step 3
Exam Tip
प्रांत पहले घटक और परिसर दूसरे घटक से बनता है। इसलिए domain ( {1,2,3} ) और range ( {4,5,6} ) हैं।
The smallest reflexive relation contains only all pairs ( (a,a) ). It is also called the identity relation.
Step 2
Why this answer is correct
The correct answer is B. ( {(1,1),(2,2),(3,3)} ). The smallest reflexive relation contains only all pairs ( (a,a) ). It is also called the identity relation.
Step 3
Exam Tip
सबसे छोटे परावर्ती संबंध में केवल सभी ( (a,a) ) युग्म होते हैं। इसे identity relation भी कहते हैं।
The largest relation is \(A\times A\) because it contains all possible ordered pairs. It is called the universal relation.
Step 2
Why this answer is correct
The correct answer is C. \( A\times A \). The largest relation is \(A\times A\) because it contains all possible ordered pairs. It is called the universal relation.
Step 3
Exam Tip
सबसे बड़ा संबंध \(A\times A\) होता है क्योंकि इसमें सभी possible ordered pairs होते हैं। इसे सार्वत्रिक संबंध कहा जाता है।
A relation on (A) is a subset of \(A\times A\), and \(|A\times A|=n^2\). Hence the total number of relations is \(2^{n^2}\).
Step 2
Why this answer is correct
The correct answer is B. \(2^{n^2}\). A relation on (A) is a subset of \(A\times A\), and \(|A\times A|=n^2\). Hence the total number of relations is \(2^{n^2}\).
Step 3
Exam Tip
(A) पर संबंध \(A\times A\) का उपसमुच्चय है और \(|A\times A|=n^2\)। इसलिए कुल संबंध \(2^{n^2}\) हैं।
In a transitive relation, ( (1,4) ) and ( (4,9) ) require ( (1,9) ). In exams, the middle element (4) is the common element.
Step 2
Why this answer is correct
The correct answer is C. \((1,9)\in R\). In a transitive relation, ( (1,4) ) and ( (4,9) ) require ( (1,9) ). In exams, the middle element (4) is the common element.
Step 3
Exam Tip
संक्रामी संबंध में ( (1,4) ) और ( (4,9) ) से ( (1,9) ) मिलना चाहिए। परीक्षा में बीच का (4) common element होता है।
Because (4+2=6), which is greater than (5), \((4,2)\notin R\). In exams, apply the condition directly to the ordered pair.
Step 2
Why this answer is correct
The correct answer is B. \((4,2)\notin R\). Because (4+2=6), which is greater than (5), \((4,2)\notin R\). In exams, apply the condition directly to the ordered pair.
Step 3
Exam Tip
क्योंकि (4+2=6), जो (5) से बड़ा है, इसलिए \((4,2)\notin R\)। परीक्षा में ordered pair पर शर्त सीधे लगाएं।
Take the first component from (A) and the second from (B), then apply (b=2a). This gives ( (1,2),(2,4),(3,6) ).
Step 2
Why this answer is correct
The correct answer is A. ( {(1,2),(2,4),(3,6)} ). Take the first component from (A) and the second from (B), then apply (b=2a). This gives ( (1,2),(2,4),(3,6) ).
Step 3
Exam Tip
पहला घटक (A) से और दूसरा (B) से लेकर (b=2a) लगाते हैं। इससे ( (1,2),(2,4),(3,6) ) मिलते हैं।
The condition gives pairs ( (3,2),(4,3),(5,4) ), so the domain is ( {3,4,5} ). In exams, take first components for domain.
Step 2
Why this answer is correct
The correct answer is B. ( {3,4,5} ). The condition gives pairs ( (3,2),(4,3),(5,4) ), so the domain is ( {3,4,5} ). In exams, take first components for domain.
Step 3
Exam Tip
शर्त से युग्म ( (3,2),(4,3),(5,4) ) बनते हैं, इसलिए प्रांत ( {3,4,5} ) है। परीक्षा में domain के लिए पहले घटक लें।
A. क्योंकि \((1,3)\notin R\)/Because \((1,3)\notin R\)
Step 1
Concept
Since \((1,2)\in R\) and \((2,3)\in R\), ( (1,3) ) is required, but it is missing. Therefore (R) is not transitive.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \((1,3)\notin R\) / Because \((1,3)\notin R\). Since \((1,2)\in R\) and \((2,3)\in R\), ( (1,3) ) is required, but it is missing. Therefore (R) is not transitive.
Step 3
Exam Tip
\((1,2)\in R\) और \((2,3)\in R\) होने पर ( (1,3) ) चाहिए, पर वह नहीं है। इसलिए (R) संक्रामी नहीं है।
B. नहीं, क्योंकि \((1,1),(2,2),(3,3)\notin R\)/No, because \((1,1),(2,2),(3,3)\notin R\)
Step 1
Concept
All pairs ( (a,a) ) are necessary for a reflexive relation. Such pairs do not occur in \(a\neq b\).
Step 2
Why this answer is correct
The correct answer is B. नहीं, क्योंकि \((1,1),(2,2),(3,3)\notin R\) / No, because \((1,1),(2,2),(3,3)\notin R\). All pairs ( (a,a) ) are necessary for a reflexive relation. Such pairs do not occur in \(a\neq b\).
Step 3
Exam Tip
परावर्ती संबंध के लिए सभी ( (a,a) ) युग्म जरूरी हैं। \(a\neq b\) में ऐसे युग्म नहीं आते।
A. हाँ, क्योंकि \(a\neq b\) होने पर \(b\neq a\) भी होता है/Yes, because if \(a\neq b\), then \(b\neq a\) also
Step 1
Concept
If two elements are different, the reversed ordered pair also has different elements. Therefore (R) is symmetric.
Step 2
Why this answer is correct
The correct answer is A. हाँ, क्योंकि \(a\neq b\) होने पर \(b\neq a\) भी होता है / Yes, because if \(a\neq b\), then \(b\neq a\) also. If two elements are different, the reversed ordered pair also has different elements. Therefore (R) is symmetric.
Step 3
Exam Tip
यदि दो अवयव अलग हैं, तो उल्टा ordered pair भी अलग अवयवों का होगा। इसलिए (R) सममित है।
A. (R) परावर्ती है लेकिन सामान्यतः सममित नहीं है/(R) is reflexive but generally not symmetric
Step 1
Concept
Since \(a\leq a\) for every (a), (R) is reflexive. But \((1,2)\in R\) while \((2,1)\notin R\), so it is not symmetric.
Step 2
Why this answer is correct
The correct answer is A. (R) परावर्ती है लेकिन सामान्यतः सममित नहीं है / (R) is reflexive but generally not symmetric. Since \(a\leq a\) for every (a), (R) is reflexive. But \((1,2)\in R\) while \((2,1)\notin R\), so it is not symmetric.
Step 3
Exam Tip
क्योंकि हर (a) के लिए \(a\leq a\) है, इसलिए (R) परावर्ती है। लेकिन \((1,2)\in R\) होने पर \((2,1)\notin R\), इसलिए यह सममित नहीं है।
