Exclusive Or with Contradiction

Theorem

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

$p \oplus \bot \vdash p$
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

$\Box$

$p \vdash p \oplus \bot$
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

$\blacksquare$

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.

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

$\blacksquare$