The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

Question

The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

Smart Achievers · Accepted Answer

(3)

Let we have set $A = {1, 2, 3}$

$By definition, we can say that$

$For reflexivity: (1, 1), (2, 2), (3, 3) \in R ...(i)$

$For symmetry: (2, 1), (1, 2) \in R ...(ii)$

$For transitivity: (1, 2) \in R and (2, 3) \in R \Rightarrow (1, 3) \in R ...(iii)$

$But according to the question,$

$For not symmetric: (2, 1) and (3, 2) \notin R ...(iv)$

$So, R_{1} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}$

$R_{2} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (2, 1)}$

$and R_{3} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (3, 2)}$

Solution

Q.
The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

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Solution

Q. The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

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Q.
The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]