Exclusive Or with Tautology

Theorem

An exclusive or with a tautology:

$p \oplus \top \dashv \vdash \neg p$

Proof by Natural Deduction

By the tableau method of natural deduction:

$p \oplus \top \vdash \neg p$
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:

$\neg p \vdash p \oplus \top$
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

$\blacksquare$

Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.

$\begin{array}{|c|ccc||cc|} \hline p & p & \oplus & \top & \neg & p \\ \hline F & F & T & T & T & F \\ T & T & F & T & F & T \\ \hline \end{array}$

$\blacksquare$