# Exclusive Or with Contradiction/Proof 1

## Theorem

$p \oplus \bot \dashv \vdash p$

## Proof

By the tableau method of natural deduction:

$p \oplus \bot \vdash p$
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:

$p \vdash p \oplus \bot$
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

$\blacksquare$