Equivalence relations with $\cap$, $\cup$ and $\circ$
Here are some hints:

a) If $E_1,E_2$ are equivalence relations then so is $E_1 \cap E_2$. The proof is just by "unpacking" the definitions. (More generally, every class defined by universal statements – those of the form $\forall x_1,\ldots,x_n \phi(x_1,\ldots,x_n)$ – is closed under intersection.)

b) When trying to prove that $E_1 \cup E_2$ is transitive, we get stuck in the following case: if $a E_1 b$ and $b E_2 c$ then we have to show that $a E_1 c$ or $a E_2 c$, though there is no reason to believe that this must be the case. This helps you construct a counterexample.

c) When trying to prove that $E_1 \circ E_2$ is symmetric, we get stuck: if $a E_1 x E_2 b$ then there is no particular reason that $b E_1 y E_2 a$ for some $y$. This helps you construct a counterexample.