Empty Set is Bottom of Lattice of Power Set

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Theorem

Let $X$ be a set.

Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be the lattice of the power set of $X$.


Then:

$\O = \bot$

where $\bot$ denotes the bottom of $L$.


Proof

By Empty Set is Subset of All Sets:

$\forall S \in \powerset X: \O \subseteq S$

By Empty Set is Element of Power Set:

$\O \in \powerset X$

Thus by definition of the smallest element:

$\O = \bot$

$\blacksquare$


Sources