Exclusive Or as Disjunction of Conjunctions
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Theorem
- $p \oplus q \dashv \vdash \paren {\neg p \land q} \lor \paren {p \land \neg q}$
Proof 1
\(\ds p \oplus q\) | \(\dashv \vdash\) | \(\ds \neg \left ({p \iff q}\right)\) | Exclusive Or is Negation of Biconditional | |||||||||||
\(\ds \) | \(\dashv \vdash\) | \(\ds \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)\) | Non-Equivalence as Disjunction of Conjunctions |
$\blacksquare$
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.
$\begin{array}{|ccc||ccccccccc|} \hline
p & \oplus & q & (\neg & p & \land & q) & \lor & (p & \land & \neg & q) \\
\hline
F & F & F & T & F & F & F & F & F & F & T & F \\
F & T & T & T & F & T & T & T & F & F & F & T \\
T & T & F & F & T & F & F & T & T & T & T & F \\
T & F & T & F & T & F & T & F & T & F & F & T \\
\hline
\end{array}$
$\blacksquare$
Sources
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 12$: Material Equivalence and Alternation