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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 $A$.

Ayman Hourieh
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mingle
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    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: http://math.stackexchange.com/questions/268726/are-there-real-life-relations-which-are-symmetric-and-reflexive-but-not-transiti/276213#276213 – alancalvitti Apr 22 '15 at 19:33

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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?

Brian M. Scott
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