Let A= 1 ,2 ,3 . The number of relations on A, containing (1 ,2 ) and (2 ,3 ), which are reflexive and transitive but not symmetric, is -

Question

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Correct answer: 3. (3) R is reflexive ⇒R have (1 ,1 ) , (2 ,2 ) , (3 ,3 ) R is transitive ∵ (1 ,2 ) , (23 ) ∈R ∴ (1 ,3 ) ∈R ∴R_1= (1 ,1 ) , (2 ,2 ) , (3 ,3 ) , (1 ,2 ) , (2 ,3 ) , (1 ,3 ) Clearly R_1 is reflexive and transitive but not symmetric. Similarly, R_2= (1 ,1 ) , (2 ,2 ) , (3 ,3 ) , (1 ,2 ) , (2 ,3 ) , (1 ,3 ) , (3 ,2 ) R_3= (1 ,1 ) , (2 ,2 ) , (3 ,3 ) , (1 ,2 ) , (2 ,3 ) , (1 ,3 ) , (2 ,1 ) Therefore, 3 relations are possible

$Let A = 1, 2, 3 . The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is -$

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