Absorption Laws (Boolean Algebras)

Theorem
Let $\struct {S, \vee, \wedge}$ be a Boolean algebra, defined as in Definition 1.

Then for all $a, b \in S$:


 * $a = a \vee \paren {a \wedge b}$
 * $a = a \wedge \paren {a \vee b}$

That is, $\vee$ absorbs $\wedge$, and $\wedge$ absorbs $\vee$.

Proof
Let $a, b \in S$.

Then:

as desired.

The result:


 * $a = a \wedge \paren {a \vee b}$

follows from the Duality Principle.