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.