Let A a b c d How many relations defined on A are reflexive

Let A = {a. b. c. d}. How many relations defined on A are reflexive, symmetric and transitive and contain the ordered pairs (a. b). (b. c). (c. d)?

Solution

Reflexive means aRa and in (a,b), (b,c), and (c,d) there are no reflexive. (You would need one of the ordered pairs (a,a), (b,b), (c,c), or (d,d)). Symmetric means if aRb then bRa so you would need (b,a), or (c,b), or (d,c) in the ordered pairs and you do not have them. Transitive means if aRb and bRc then aRc. If you had transitivity then one of the following would have to be in the set of ordered pairs: aRb and bRc then you need aRc and you don\'t. Likewise if bRc and cRd then you need bRd and you don\'t have that either.
So there is no relation in the set A = [a,b,c,d} of reflexive, symmetric, or transitive with the ordered pairs (a,b), (b,c), and (c,d).