Exclusive Or as Conjunction of Disjunctions

Theorem

$p \oplus q \dashv \vdash \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)$

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