Exclusive Or with Tautology/Proof 1
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Theorem
- $p \oplus \top \dashv \vdash \neg p$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \oplus \top$ | Premise | (None) | ||
| 2 | 1 | $\left({p \lor \top} \right) \land \neg \left({p \land \top}\right)$ | Sequent Introduction | 1 | Definition of Exclusive Or | |
| 3 | 1 | $\top \land \neg \left({p \land \top}\right)$ | Sequent Introduction | 1 | Disjunction with Tautology | |
| 4 | 1 | $\neg \left({p \land \top}\right)$ | Sequent Introduction | 1 | Conjunction with Tautology | |
| 5 | 1 | $\neg p$ | Sequent Introduction | 1 | Conjunction with Tautology |
$\Box$
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\neg p$ | Assumption | (None) | ||
| 2 | $\top$ | Rule of Top-Introduction: $\top \II$ | (None) | |||
| 3 | $p \lor \top$ | Sequent Introduction | 2 | Disjunction with Tautology | ||
| 4 | 1 | $\neg \left({p \land \top}\right)$ | Sequent Introduction | 1 | Conjunction with Tautology | |
| 5 | 1 | $\left({p \lor \top}\right) \land \neg \left({p \land \top}\right)$ | Rule of Conjunction: $\land \II$ | 3, 4 | ||
| 6 | 1 | $p \oplus \top$ | Sequent Introduction | 5 | Definition of Exclusive Or |
$\blacksquare$