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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): atom
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): atom
- Mizar article YELLOW_1:18