Complement of Top/Boolean Algebra

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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


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