Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Proof 2

From ProofWiki
Jump to navigation Jump to search

Theorem

$p \lor \left ({p \land q}\right) \dashv \vdash p$


Proof

\(\displaystyle p \lor \left({p \land q}\right)\) \(=\) \(\displaystyle \left({p \land \top}\right) \lor \left({p \land q}\right)\) Conjunction with Tautology
\(\displaystyle \) \(=\) \(\displaystyle p \land \left({\top \lor q}\right)\) Conjunction is Left Distributive over Disjunction
\(\displaystyle \) \(=\) \(\displaystyle p \land \top\) Disjunction with Tautology
\(\displaystyle \) \(=\) \(\displaystyle p\) Conjunction with Tautology

$\blacksquare$