# Precisely One Function in terms of And, Or and Not

## Theorem

Let $\map P {A, B, C}$ denote the precisely one function on the statements $A$, $B$ and $C$.

Then:

$\map P {A, B, C} \dashv \vdash \paren {A \land \neg B \land \neg C} \lor \paren {\neg A \land B \land \neg C} \lor \paren {\neg A \land \neg B \land C}$

where:

$\land$ denotes conjunction
$\lor$ denotes disjunction
$\neg$ denotes negation

## Proof

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.

$\begin{array}{|cccc||ccccc|} \hline P & (A & B & C) & (((A & \land & \neg & B) & \land & \neg & C) & \lor & ((\neg & A & \land & B) & \land & \neg & C)) & \lor & ((\neg & A & \land & \neg & B) & \land & C) \\ \hline F & F & F & F & F & F & T & F & F & T & F & F & T & F & F & F & F & T & F & F & T & F & T & T & F & F & F \\ T & F & F & T & F & F & T & F & F & F & T & F & T & F & F & F & F & F & T & T & T & F & T & T & F & T & T \\ T & F & T & F & F & F & F & T & F & T & F & T & T & F & T & T & T & T & F & T & T & F & F & F & T & F & F \\ F & F & T & T & F & F & F & T & F & F & T & F & T & F & T & T & F & F & T & F & T & F & F & F & T & F & T \\ T & T & F & F & T & T & T & F & T & T & F & T & F & T & F & F & F & T & F & T & F & T & F & T & F & F & F \\ F & T & F & T & T & T & T & F & F & F & T & F & F & T & F & F & F & F & T & F & F & T & F & T & F & F & T \\ F & T & T & F & T & F & F & T & F & T & F & F & F & T & F & T & F & T & F & F & F & T & F & F & T & F & F \\ F & T & T & T & T & F & F & T & F & F & T & F & F & T & F & T & F & F & T & F & F & T & F & F & T & F & T \\ \hline \end{array}$

$\blacksquare$