Let A={1,2,3},then find number of relation containing(1,2)and(2,3)which are reflexive and transitive but not symmetric.

The smallest reflexive relation on set A containing (1, 2) and (2, 3) is

  

R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}

  

Since (1, 2) belongs to R and (2, 3) belongs to R but (1, 3) does not belong to R. So, R is not transitive. To make it transitive , include (1, 3) in R.

  

R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}

 

This is reflexive and transitive but not symmetric.

 

Now, if we add the pair (2, 1) to R1 to get R2, thwn the relation R2 is still reflexive, transitive but not symmetric. Similarly, by adding (3,2) and (3,1) respectively to R1,

 

R3 ={(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (3,2)}

R4 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (3,2), (3, 1)}

 

These realtions are reflexive and transitive but not symmetric.

 

We observe that out of ordered pairs (2,1), (3,2) and (3,1) at a time if we add any two oredered pairs at a time to R1, then to maintain the transitivity we will be forced to add the remaining third pair and in this process the relation will become symmetric also which is not required. Hence, total number of reflexive, transitive but not symmetric relations containing (1,2) and (2,3) is three.