De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 2
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Theorem
- $\vdash \paren {\neg p \lor \neg q} \iff \paren {\neg \paren {p \land q} }$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\neg p \lor \neg q$ | Assumption | (None) | ||
| 2 | 1 | $\neg \paren {p \land q}$ | Sequent Introduction | 1 | De Morgan's Laws (Logic): Disjunction of Negations: Formulation 1 | |
| 3 | $\paren {\neg p \lor \neg q} \implies \paren {\neg \paren {p \land q} }$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
| 4 | 4 | $\neg \paren {p \land q}$ | Assumption | (None) | ||
| 5 | 4 | $\neg p \lor \neg q$ | Sequent Introduction | 4 | De Morgan's Laws (Logic): Disjunction of Negations: Formulation 1 | |
| 6 | $\paren {\neg \paren {p \land q} } \implies \paren {\neg p \lor \neg q}$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
| 7 | $\paren {\neg p \lor \neg q} \iff \paren {\neg \paren {p \land q} }$ | Biconditional Introduction: $\iff \II$ | 3, 6 |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.
$\begin{array}{|ccccc|c|cccc|} \hline \neg & p & \lor & \neg & q & \iff & \neg & (p & \land & q) \\ \hline \T & \F & \T & \T & \F & \T & \T & \F & \F & \F \\ \T & \F & \T & \F & \T & \T & \T & \F & \F & \T \\ \F & \T & \T & \T & \F & \T & \T & \T & \F & \F \\ \F & \T & \F & \F & \T & \T & \F & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T65}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $10.$
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $13$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(2) \ \text{(iv)}$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.14$: Exercise $12 \ (10)$
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if... $\text{(c)}$
- (referring to it as one of the laws of negation)