How can I show that this is a partial order?

Question

Let $S$ be an arbitrary amount and define the relation $R \subseteq \mathcal{P}(S) \times \mathcal{P}(S)$ so that $(A,B) \in R$ if and only if $A \supseteq B$. Here $\mathcal{P}(S)$ is a spelling for the power set to $S$. Show that $(\mathcal{P}(S),R)$ is a partially ordered quantity.

Answer 1

Hint: You must show the following for all $A,B,C\in\mathcal{P}(S)$:

(1) $A\supseteq A$--that is, $R$ is reflexive on $\mathcal{P}(S)$.

(2) If $A\supseteq B$ and $B\supseteq C$, then $A\supseteq C$--that is, $R$ is transitive on $\mathcal{P}(S)$.

(3) If $A\supseteq B$ and $B\supseteq A$, then $A=B$--that is, $R$ is antisymmetric on $\mathcal{P}(S)$.

By definition, $A\supseteq B$ if and only if for all $x\in B$ we have $x\in A$. Two sets $A$ and $B$ are equal if and only if they have all the same elements. From there, the proofs are straightforward.