Is the relation "x is a subset of y" on the power set $P(S)$ of a finite set $S$ a lattice? My professor asked this question.
We defined in class , a lattice , a set of elements, where every pair of them has an infinum and a supremum. For example if: $S =$ {$1,2,...,18$} and $x = ${$2,3 , 4$} while $y = ${$2,3,4,5$}. What I do not get , is how to apply the definition of infinum and supremum in the relation "$ \subseteq$". For example, if there was such z , that was an infinum for x,y should it be a subset with less (in quantity elements of S ) like {2,3} ? What exactly do we compare , to get a result here?
For an arbitrary set $S$, the power set $\mathcal P(S)$ is a lattice under the ordering $\subseteq$ using $\cap$ as inf and $\cup$ as sup.
To prove this, you just need to show the following:
$$\forall U,V \in \mathcal P(S), U \cap V \subseteq U \land U \cap V \subseteq V$$ $$\forall U,V,W \in \mathcal P(S), W \subseteq U \land W \subseteq V \implies W \subseteq U \cap V$$ $$\forall U,V \in \mathcal P(S), U \cup V \supseteq U \land U \cup V \supseteq V$$ $$\forall U,V,W \in \mathcal P(S), W \supseteq U \land W \supseteq V \implies W \supseteq U \cup V$$
Note that statements (3) and (4) are just obtained from (1) and (2) by taking $\cap \to \cup$ and $\subseteq \to \supseteq$.
You should be able to prove these statements to verify that $\mathcal P(S)$ is indeed a lattice.