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.
Sources
- 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.1$: Boolean operators
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.4.1$