De Morgan's Laws (Logic)/Conjunction/Formulation 1/Forward Implication
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Theorem
- $p \land q \vdash \neg \left({\neg p \lor \neg q}\right)$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \land q$ | Premise | (None) | ||
| 2 | 1 | $p$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
| 3 | 1 | $q$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
| 4 | 4 | $\neg p \lor \neg q$ | Assumption | (None) | ||
| 5 | 5 | $\neg p$ | Assumption | (None) | ||
| 6 | 1, 5 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 2, 5 | ||
| 7 | 7 | $\neg q$ | Assumption | (None) | ||
| 8 | 1, 7 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 3, 7 | ||
| 9 | 1, 4 | $\bot$ | Proof by Cases: $\text{PBC}$ | 4, 5 – 6, 7 – 8 | Assumptions 5 and 7 have been discharged | |
| 10 | 1 | $\neg \left({\neg p \lor \neg q}\right)$ | Proof by Contradiction: $\neg \II$ | 4 – 9 | Assumption 4 has been discharged |
$\blacksquare$
Sources
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction: Exercises $1.6: \ 2 \ \text{(a)}$