Three-Way Exclusive Or and Equivalence

Theorem

Let $p \iff q$ be the equivalence operation, and $p \oplus q$ be the exclusive or operation.

Then:

$p \iff q \iff r \dashv \vdash p \oplus q \oplus r$

Proof

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 under the main connectives match for all models.

$\begin{array}{|ccccc||ccccc|} \hline (p & \iff & q) & \iff & r & (p & \oplus & q) & \oplus & r \\ \hline F & T & F & F & F & F & F & F & F & F \\ F & T & F & T & T & F & F & F & T & T \\ F & F & T & T & F & F & T & T & T & F \\ F & F & T & F & T & F & T & T & F & T \\ T & F & F & T & F & T & T & F & T & F \\ T & F & F & F & T & T & T & F & F & T \\ T & T & T & F & F & T & F & T & F & F \\ T & T & T & T & T & T & F & T & T & T \\ \hline \end{array}$

$\blacksquare$

From Equivalence Properties and Exclusive Or Properties, we have that both $\iff$ and $\oplus$ are associative, which justifies the rendition of this result without parentheses.

Comment

A bizarrely non-intuitive result which is rarely documented.