# De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1/Forward Implication

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## Theorem

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

## Proof

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\neg p \land \neg q$ | Premise | (None) | ||

2 | 1 | $\neg p$ | Rule of Simplification: $\land \EE_1$ | 1 | ||

3 | 1 | $\neg q$ | Rule of Simplification: $\land \EE_2$ | 1 | ||

4 | 4 | $p \lor q$ | Assumption | (None) | ||

5 | 5 | $p$ | Assumption | (None) | ||

6 | 1, 5 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 5, 2 | ||

7 | 7 | $q$ | Assumption | (None) | ||

8 | 1, 7 | $\bot$ | Principle of Non-Contradiction: $\neg \EE$ | 7, 3 | ||

9 | 1, 4 | $\bot$ | Proof by Cases: $\text{PBC}$ | 4, 5 – 6, 7 – 8 | Assumptions 5 and 7 have been discharged | |

10 | 1 | $\neg \paren {p \lor q}$ | 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): $\S 1.2.1$: Rules for natural deduction: Exercises $1.5: \ 2 \ \text{(a)}$