Power Set is Lattice

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Theorem

Let $S$ be a set.

Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the subset relation $\subseteq$.

Then $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is a lattice.


Proof

From Subset Relation on Power Set is Partial Ordering, we have that $\subseteq$ is a partial ordering.

Let $X, Y \in \mathcal P \left({S}\right)$.

Then from Union is Smallest Superset:

$X \subseteq T, Y \subseteq T \iff X \cup Y \subseteq T$

and from Intersection is Largest Subset:

$X \subseteq T, Y \subseteq T \iff T \subseteq X \cap Y$

So $X \cap Y$ is the infimum and $X \cup Y$ is the supremum of $\left\{{X, Y}\right\}$.

Hence by definition $\mathcal P \left({S}\right)$ is a lattice.

$\blacksquare$


Also see


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