Complement of Bottom/Boolean Algebra
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Theorem
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra.
Then:
- $\neg \bot = \top$
Proof
Since $\bot$ is the identity for $\vee$, the first condition for $\neg \bot$:
- $\bot \vee \neg \bot = \top$
implies that $\neg \bot = \top$ is the only possibility.
Since $\top$ is the identity for $\wedge$, it follows that:
- $\bot \wedge \top = \bot$
and we conclude that:
- $\neg \bot = \top$
as desired.
$\blacksquare$
Also see
Sources
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 2$: Exercise $2$