Complement of Top/Boolean Algebra

Theorem

Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra.

Then $\neg \top = \bot$.

Proof

Since $\top$ is the identity for $\wedge$, the second condition for $\neg \top$:

$\top \wedge \neg \top = \bot$

implies that $\neg \top = \bot$ is the only possibility.

Since $\bot$ is the identity for $\vee$, it follows that:

$\top \vee \bot = \top$

and we conclude that:

$\neg \top = \bot$

as desired.

$\blacksquare$

Also see

• Complement of Bottom (Boolean Algebras)

Sources

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