Properties of Binary Relations (Specifically, transitivity)

Question

Suppose there is a set A = {a, b, c}. A binary relation on A is R0 = {(a, a), (b, b), (c, c)}.

I have been told that R0 is a preorder of A but am not seeing how this is possible. How is it transitive? Say you label (a, a) as (x, y) and (b, b) as (y, z). If transitivity were upheld, wouldn't (a, b) have to be present in this binary relation to satisfy the presence of (x, z)?

Answer 1

Your relation is transitive because it is valid only on pairs with members that are both a or both b or both c.

That means if you have (a1, a2) and (a2, a3) where all a1, a2 and a3 are a - you will have (a1, a3) because both a1 and a3 are a.

Your example is not valid because if you have (x, y) which is indeed (a, a) and also you have (y, z) which is in turn (b, b) - it will be required that y is both a and b at the same time, in other words it will be required that a is b but that is false because a and b are different (actually can be different) - according to definition of problem conditions.