Let $X$ be the set ${\{a,b,d,e,f}\}$, and let $R$ be the relation ${\{(a,a),(a,d),(a,e),(b,b),(b,c),(c,c),(c,b),(d,d_, (d,e),(e,e)}\}$.
Now, I'm trying to find a definable subset of $X$ which satisfies $\exists u(R(u,v) \land u \neq v)$.
I have no idea what this formula means, though; I know $R(u,v)$ means that $u$ and $v$ are related; thus there is a link from $u$ to $v$ on the directed graph but what does $R(u,v) \land u$ even mean?
I believe it asks to find the subset consisting of all elements $v$ such that $\exists u(R(u,v) \land u \neq v)$
So, if we call that subset $S$, we can define $S$ as: $S = \{ v \in X | \exists u (R(u,v) \land u \not = v \}$
In other words: find all element such that some different element is related to it.
Thus, for example, $d \in S$, since $a$ is related to it, i.e. $(a,d) \in R$.
But nothing is related to $a$ other than $a$ itself, and that is not different from $a$, so $a \not \in S$.