Absorption Laws (Logic)/Conjunction Absorbs Disjunction
Jump to navigation
Jump to search
Theorem
- $p \land \paren {p \lor q} \dashv \vdash p$
This can be expressed as two separate theorems:
Forward Implication
- $p \land \left({p \lor q}\right) \vdash p$
Reverse Implication
- $p \vdash p \land \left({p \lor q}\right)$
Proof 1
We apply the Method of Truth Tables.
As can be seen by inspection, the appropriate truth values match for all boolean interpretations.
$\begin{array}{|ccccc||c|} \hline p & \land & (p & \lor & q) & p \\ \hline F & F & F & F & F & F \\ F & F & F & T & T & F \\ T & T & T & T & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$
$\blacksquare$
Proof 2
By calculation:
\(\ds p \land \left({p \lor q}\right)\) | \(=\) | \(\ds \left({p \lor \bot}\right) \land \left({p \lor q}\right)\) | Disjunction with Contradiction | |||||||||||
\(\ds \) | \(=\) | \(\ds p \lor \left({\bot \land q}\right)\) | Disjunction is Left Distributive over Conjunction | |||||||||||
\(\ds \) | \(=\) | \(\ds p \lor \bot\) | Conjunction with Contradiction | |||||||||||
\(\ds \) | \(=\) | \(\ds p\) | Disjunction with Contradiction |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 1.5$: Further Proofs: Résumé of Rules: Theorem $31$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.5$