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