Complement of Bottom/Boolean Algebra

Theorem
Let $\left({S, \vee, \wedge, \neg}\right)$ 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.

Also see

 * Complement of Top