question_answer 73)

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is (a) 1 (b) 2 (c) 3 (d) 4  

Answer:

Given A = {1, 2, 3} and 1 is related to 2. Case ? I : WhenI is related to 3.                   As (1, 1), (2, 2) and (3, 3) belongs to R1              is reflexive.       Also        is symmetric.       Further a R1b and bR1 c        is transitive.       Hence R2 is an equivalence relation.       Case-II : When 1 is not related to 3.                         As (1, 1), (2, 2), (3, 3) belongs to R2              is reflexive.       Now (1, 2) and (2, 1) belongs to R2             is symmetric.       Also aRb and bRc        is transitive.       Hence R2 is an equivalenct relation.       Thus there are two equivalence relations.       Hence (b) is correct.