How to define a compatible relation using a cover

A binary relation on a set that is reflexive and symmetric is called a compatible relation. Let $A$ be a set. A cover of $A$ is a set of non empty subsets of $A$, say $\{ A_1, A_2, A_3 \ldots, A_n\}$ such that union of $A_i$'s is equal to $A$. Suggest a way to define a compatible relation on $A$ from a cover of $A$.

• If you show us how you have tried to answer this question perhaps someone can use your work to give a good hint or answer this question. – Jay Sep 28 '13 at 20:38
• Binary reflexive, symmetric relations are also called tolerances: math.stackexchange.com/questions/268726/… – alancalvitti Apr 22 '15 at 19:33

HINT: Let $\mathscr{C}$ be a cover of $A$. For each $a\in A$ let $\mathscr{C}_a=\{C\in\mathscr{C}:a\in C\}$. Use the sets $\mathscr{C}_a$ to define a compatible relation $\sim$ on $A$: for $a,b\in A$, $a\sim b$ if and only if $\mathscr{C}_a$ and $\mathscr{C}_b$ are ... what?