Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Forward Implication
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Theorem
- $p \lor \paren {p \land q} \vdash p$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \lor \paren {p \land q}$ | Premise | (None) | ||
| 2 | 2 | $p$ | Assumption | (None) | ||
| 3 | 3 | $p \land q$ | Assumption | (None) | ||
| 4 | 3 | $p$ | Rule of Simplification: $\land \EE_1$ | 3 | ||
| 5 | 1 | $p$ | Proof by Cases: $\text{PBC}$ | 1, 2 – 2, 3 – 4 | Assumptions 2 and 3 have been discharged |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $5$ Further Proofs: Résumé of Rules: Theorem $32 \ \text{(a)}$
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Exercises $1.4: \ 2 \ \text{(n)}$