Absorption Laws (Logic)/Disjunction Absorbs Conjunction

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Theorem

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


This can be expressed as two separate theorems:

Forward Implication

$p \lor \paren {p \land q} \vdash p$

Reverse Implication

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


Proof 1

We apply the Method of Truth Tables.

As can be seen by inspection, the appropriate truth values match for all boolean interpretations.

$\begin{array}{|ccccc||c|} \hline p & \lor & (p & \land & q) & p \\ \hline F & F & F & F & F & F \\ F & F & F & F & T & F \\ T & T & T & F & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$


Proof 2

\(\displaystyle p \lor \left({p \land q}\right)\) \(=\) \(\displaystyle \left({p \land \top}\right) \lor \left({p \land q}\right)\) 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

$\blacksquare$


Also see


Sources