Complement of Bottom/Boolean Algebra

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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.

$\blacksquare$


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