Would this relation be an equivalence relation? I am a bit stuck on this one question from my homework and for some reason it isn't making any sense to me. I would really appreciate it if somebody could explain it to me how I can go about to solving it.
Suppose that $\sim_1$ and $\sim_2$ are equivalence relations on the set $S$. Define a new relation $\sim_v$ so that
$$a \sim_v b \Leftrightarrow a\sim_1 b \vee a\sim_2 b.$$
Is $\sim_v$ always an equivalence relation? (Specifically, it is true that for every pair of equivalence relations $\sim_1$ and $\sim_2$ the relation $\sim_v$ is an equivalence relation?) If so, prove it. If not, give an example where it fails and explain why it fails for your example.

Define $\sim_1$ on the set of integers such that $a\sim_1 b$ if and only if $b-a$ is divisible by $2$.
Define $\sim_2$ on the set of integers such that $a\sim_2 b$ if and only if $b-a$ is divisible by $3$.
Define $\sim_v$ on the set of integers such that $a\sim_v b$ if and only if $a\sim_1 b$ or $a\sim_2 b$.
Verify that the first two are equivalence relations but the last is not as it is not transitive.
