1)Prove or disprove: If A is a set for which each $\displaystyle R \in A$ is a transitive relation, then $\displaystyle \bigcup A$ is a transitive relation.

I think this is true. So I tried to prove it. Let $\displaystyle (x,y) \in \bigcup A$ and $\displaystyle (y,z) \in \bigcup A$. Then by definition of union, there exists $\displaystyle R$ such that $\displaystyle (x,y) \in R \in A$ and $\displaystyle (y,z) \in R \in A$. Since R is transitive by definition of set A, we have $\displaystyle (x,z) \in R \in A$. Hence, $\displaystyle (x,z) \in \bigcup A$

I am not sure about my argument here. Is it possible that $\displaystyle (x,y) \in R \in A$ while $\displaystyle (y,z) \in R' \in A$?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 $\displaystyle \bigcup A$ 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 $\displaystyle \bigcap A$, and it seems that three properties of equivalence relation do hold for this case, but for union, somehow I don't see it follows.