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.