Absorption Laws (Logic)

Theorem

 * $$p \and \left({p \or q}\right) \dashv \vdash p$$
 * $$p \or \left ({p \and q}\right) \dashv \vdash p$$

These are called the Absorption Laws or Absorption Identities.

Proof by Natural deduction
These are proved by the Tableau method.

Proof by Truth Table
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

We see that $$v \left({p}\right) = v \left({p \and \left({p \or q}\right)}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.

We see that $$v \left({p}\right) = v \left({p \or \left({p \and q}\right)}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.