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Let $S$ be a set that contains at least two different elements. Let $R$ be the relation on $P(S)$, the power set of $S$, defined by $(X,Y)\in R$ if and only if $X\oplus Y = \varnothing$. Determine whether R is reflexive, symmetric, anti-symmetric, or transitive.

I'm having trouble just setting up the question. Is it $\{x,y,(x,y)\}$?

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    Hint. Write down all $8$ subsets of the set $\{1,2,3\}$. Then think about when pairs of those subsets have an empty symmetric difference. – 2017-02-19
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    They have an empty symmetric difference when they both the same right? Does that make it anti-symmetric? – 2017-02-19
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    Yes, the symmetric difference of a set with itself is empty. That tells you the relation is reflexive: every set is related to itself. There are of course other ways the symmetric difference can be empty. You should write out much of the example I suggested in my comment, not ask a question at a time in further comments. You can edit your question to show your progress and ask where you're stuck. – 2017-02-19

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