Thread: Transitive relation

1)Prove or disprove: If A is a set for which each is a transitive relation, then is a transitive relation.

I think this is true. So I tried to prove it. Let and . Then by definition of union, there exists such that and . Since R is transitive by definition of set A, we have . Hence,
I am not sure about my argument here. Is it possible that while ?Any help is appreciated.

2) Prove or disprove: If A is a nonempty set each of whose elements is an equivalence relation on the set B, then is an equivalence relation on B.
I don't think this is correct, but I can't come up with a counterexample. I tried the problem but with , and it seems that three properties of equivalence relation do hold for this case, but for union, somehow I don't see it follows.

Hi
1) Check

Then by definition of union, there exists such that and

Take care, as you said after, the "first " has no to be equal to the "second one"!

2) Here again, think of a particular but simple case where has only two elements. What if there is one class in the first relation of equivalence intersects partially a class of the second relation, for instance: (the idea is to lose the transitivity)