Meet with Complement is Bottom

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Theorem

Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 2.


Then:

$\exists \bot \in S: \forall a \in S: a \wedge \neg a = \bot$

where $\wedge$ denotes the meet operation in $S$.


This element $\bot$ is unique for any given $S$, and is named bottom.


Proof

Let $\exists r, s \in S: r \wedge \neg r = a, \ s \wedge \neg s = b$

Then:

\(\displaystyle a\) \(=\) \(\displaystyle r \wedge \neg r\) by hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \paren {s \wedge \neg s} \vee \paren {r \wedge \neg r}\) Boolean Algebra: Axiom $(BA_2 \ 5)$
\(\displaystyle \) \(=\) \(\displaystyle \paren {r \wedge \neg r} \vee \paren {s \wedge \neg s}\) Boolean Algebra: Axiom $(BA_2 \ 1)$
\(\displaystyle \) \(=\) \(\displaystyle s \wedge \neg s\) Boolean Algebra: Axiom $(BA_2 \ 5)$
\(\displaystyle \) \(=\) \(\displaystyle b\) by hypothesis


Thus, whatever $r$ and $s$ may be:

$r \wedge \neg r = s \wedge \neg s$

This unique element can be assigned the symbol $\bot$ and named bottom as required.

$\blacksquare$


Sources