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.

Also see

 * Complement of Bottom (Boolean Algebras)