# Operations of Boolean Algebra are Idempotent

## Definition

Let $\struct {S, \vee, \wedge}$ be a Boolean algebra.

Then:

$\forall x \in S: x \wedge x = x = x \vee x$

That is, both $\vee$ and $\wedge$ are idempotent operations.

## Proof

Let $x \in S$.

Then:

 $\ds x$ $=$ $\ds x \vee \bot$ as $\bot$ is the identity of $\vee$ $\ds$ $=$ $\ds x \vee \paren {x \wedge \neg x}$ as $x \wedge \neg x = \bot$ $\ds$ $=$ $\ds \paren {x \vee x} \wedge \paren {x \vee \neg x}$ both $\vee$ and $*$ distribute over the other $\ds$ $=$ $\ds \paren {x \vee x} \wedge \top$ as $x \vee \neg x = \top$ $\ds$ $=$ $\ds x \vee x$

So $x = x \vee x$.

$\Box$

The result $x = x \wedge x$ follows from Duality Principle (Boolean Algebras).

$\blacksquare$