Help with set theory question about binary relations

Question

On my assignment I was asked the question:

Determine, with reason, if the binary relation is reflexive, symmetric, antisymmetric, or transitive.

Let X be any set containing at least three distinct elements a, b, c ∈ X. Let S be the relation on P(X) such that (A, B) ∈ S when A ∩ B = {a}.

I was wondering if anyone can point me in the right direction here as I don't even know where to begin. Where do the A and B come from? If S is the relation on P(X) then does it mean that (A,B) is only in the relation if its intersection returns only the element {a}? I can't seem to wrap my head around this question, any help would be appreciated. Thank you all for taking the time to read this.

Answer 1

That $S$ is a relation on $P(X)$ mean that $A$ and $B$ are subsets of $X$. So for any two sets $A,B\subseteq X$ we say the relation $S(A,B)$ hold, or in other words $(A,B)\in S$ if $A\cap B=\{a\}$. Some hints to look at the properties:

1. Reflexivity: R is reflexive if for each $A\subseteq X$ it hold that $(A,A)\in S$ i.e. $A\cap A = \{a\}$, does this hold?
2. Symetry: R is symetric if for each $A,B\subseteq X$, such that $(A,B)\in S$ it hold that $(B,A)\in S$. If you translate $(A,B)\in S$ to the definition of S, then does it hold?
3. Anti-symetry: If $A,B \subseteq X$ and both $(A,B),(B,A)\in S$ then $A=B$. If you translate $(A,B),(B,A)\in S$ by the definition, then does this hold?
4. Transitivity: If $A,B,C\subseteq X$ and $(A,B),(B,C)\in S$ then $(A,C)\in S$. Does this hold?