Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Reverse Implication
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Theorem
- $p \vdash p \land \paren {p \lor q}$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p$ | Premise | (None) | ||
| 2 | 1 | $p \lor q$ | Rule of Addition: $\lor \II_1$ | 1 | ||
| 3 | 1 | $p \land \paren {p \lor q}$ | Rule of Conjunction: $\land \II$ | 1, 2 |
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $5$ Further Proofs: Résumé of Rules: Theorems $31 \ \text{(b)}$