Find non empty $X$ and $Y$ that satisfy the following: $\forall x \in X \;\exists y\in Y p(x,y) \;\; and \;\; \forall y \in Y \; \exists x\in X \; not \; p(x,y)$, where $p(x,y)$ is defined as the propositional function $x-y < 10$.
I cannot think of an $X$ and $Y$ that could satisfy such a condition. If for all $y$ in $Y$ there exists some $x$ in $X$ such that $not \; p(x,y)$, then it must be the case that there exists some $x \in X$ such that for all $y \in Y$ not $p(x,y)$, no? But, this is the negation of the first conjunct.
Let both $X$ and $Y$ be the set of all real numbers. Check that this works. Then go back to your reasoning for why such $X$ and $Y$ lead to a contradiction, and pinpoint where that reasoning goes wrong in this example. (This pinpointing is not strictly required by the question, but I think it's important to prevent similar errors in future questions.)