Exclusive Or with Tautology
Jump to navigation
Jump to search
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:
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 |
$\blacksquare$
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.
$\begin{array}{|c|ccc||cc|} \hline p & p & \oplus & \top & \neg & p \\ \hline F & F & T & T & T & F \\ T & T & F & T & F & T \\ \hline \end{array}$
$\blacksquare$
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$