Exclusive Or as Conjunction of Disjunctions

Theorem

$p \oplus q \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$


Proof 1

\(\ds p \oplus q\) \(\dashv \vdash\) \(\ds \paren {p \lor q} \land \neg \paren {p \land q}\) Definition of Exclusive Or
\(\ds \) \(\dashv \vdash\) \(\ds \paren {p \lor q} \land \paren {\neg p \lor \neg q}\) De Morgan's Laws: Disjunction of Negations

$\blacksquare$


Proof 2

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.


$\begin{array}{|ccc||ccccccccc|} \hline p & \oplus & q & (p & \lor & q) & \land & (\neg & p & \lor & \neg & q) \\ \hline \F & \F & \F & \F & \F & \F & \F & \T & \F & \T & \T & \F \\ \F & \T & \T & \F & \T & \T & \T & \T & \F & \T & \F & \T \\ \T & \T & \F & \T & \T & \F & \T & \F & \T & \T & \T & \F \\ \T & \F & \T & \T & \T & \T & \F & \F & \T & \F & \F & \T \\ \hline \end{array}$

$\blacksquare$


Sources

  • 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants
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