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

## Theorem

- $\neg p \lor \neg q \vdash \neg \left({p \land q}\right)$

## Proof

By the tableau method of natural deduction:

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

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

2 | 2 | $p \land q$ | Assumption | (None) | ||

3 | 2 | $p$ | Rule of Simplification: $\land \mathcal E_1$ | 2 | ||

4 | 2 | $q$ | Rule of Simplification: $\land \mathcal E_2$ | 2 | ||

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

6 | 2, 5 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 3, 5 | ||

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

8 | 2, 7 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 4, 7 | ||

9 | 1, 2 | $\bot$ | Proof by Cases: $\text{PBC}$ | 1, 5 – 6, 7 – 8 | Assumptions 5 and 7 have been discharged | |

10 | 1 | $\neg \left({p \land q}\right)$ | Proof by Contradiction: $\neg \mathcal I$ | 2 – 9 | Assumption 2 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 $1.5: \ 1 \ \text{(b)}$