# Unary Truth Functions

## Theorem

There are $4$ distinct unary truth functions:

$(1): \quad$ The constant function $\map f p = \F$
$(2): \quad$ The constant function $\map f p = \T$
$(3): \quad$ The identity function $\map f p = p$
$(4): \quad$ The logical not function $\map f p = \neg p$

## Proof

From Count of Truth Functions there are $2^{\paren {2^1} } = 4$ distinct truth functions on $1$ variable.

These can be depicted in a truth table as follows:

$\begin{array}{|c|cccc|} \hline p & \circ_1 & \circ_2 & \circ_3 & \circ_4 \\ \hline \T & \T & \T & \F & \F \\ \F & \T & \F & \T & \F \\ \hline \end{array}$

$\circ_1$: Whether $p = \T$ or $p = \F$, $\map {\circ_1} p = \T$.

Thus $\circ_1$ is the constant function $\map {\circ_1} p = \T$.

$\circ_2$: We have:

$(1): \quad p = \T \implies \map {\circ_2} p = \T$
$(2): \quad p = \F \implies \map {\circ_2} p = \F$

Thus $\circ_2$ is the identity function $\map {\circ_2} p = p$.

$\circ_3$: We have:

$(1): \quad p = \T \implies \map {\circ_3} p = \F$
$(2): \quad p = \F \implies \map {\circ_3} p = \T$

Thus $\circ_3$ is the logical not function $\map {\circ_3} p = \neg p$.

$\circ_4$: Whether $p = \T$ or $p = \F$, $\map {\circ_4} p = \F$.

Thus $\circ_4$ is the constant function $\map {\circ_4} p = \F$.

All four have been examined, and there are no other unary truth functions.

$\blacksquare$

## Linguistic Note

The word unary is pronounced yoo-nary.

Hence when the indefinite article precedes it, the form is (for example) a unary operation.