Determine if the binary relation is reflexive, symmetric, anti-symmetric, or transitive.

Question

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

I'm not even sure how to write out this relation.

Here's a start: $S = \left\{(a,b)\in \mathbb{P}(X) \times \mathbb{P}(X)\mid A\cap B=\{a\}\right\}$.

Unfortunately that doesn't make any sense to me.

Answer 1

That's almost right, except it should say $(A,B)$ instead of $(a,b)$.

Answer 2

The relation $S$ is not reflexive because $(\{b\};\{b\})$ is not in $S$, it is symmetric because $B\cap A=A\cap B$, it is not transitive because if $A=\{a,b\}$ and $B=\{a,c\}$ then $(A,B)\in S$ and $(B,A)\in S$ but $(A,A)$ is not in $S$, it is not antisymmetric as shown in the above example we get not $A=B$ .