Complement of Bottom/Boolean Algebra

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

• Complement of Top (Boolean Algebras)

Sources

• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 2$: Exercise $2$