Binary Truth Functions

Theorem
There are $16$ distinct binary truth functions:


 * Two constant functions:
 * $\map {f_F} {p, q} = F$
 * $\map {f_T} {p, q} = T$


 * Two projections:
 * $\map {\pr_1} {p, q} = p$
 * $\map {\pr_2} {p, q} = q$


 * Two negated projections:
 * $\map {\overline {\pr_1} } {p, q} = \neg p$
 * $\map {\overline {\pr_2} } {p, q} = \neg q$


 * The conjunction: $p \land q$
 * The disjunction: $p \lor q$


 * Two conditionals:
 * $p \implies q$
 * $q \implies p$


 * The biconditional: $p \iff q$
 * The exclusive or: $\map \neg {p \iff q}$


 * Two negated conditionals:
 * $\map \neg {p \implies q}$
 * $\map \neg {q \implies p}$


 * The NAND: $p \uparrow q$
 * The NOR: $p \downarrow q$

Proof
From Count of Truth Functions there are $2^{\paren {2^2} } = 16$ distinct truth functions on $2$ variables.

These can be depicted in a truth table as follows:

$\begin{array}{|r|cccc|} \hline p                               & T & T & F & F \\ q                               & T & F & T & F \\ \hline \map {f_T} {p, q}               & T & T & T & T \\ p \lor q                        & T & T & T & F \\ p \impliedby q                  & T & T & F & T \\ \map {\pr_1} {p, q}             & T & T & F & F \\ p \implies q                    & T & F & T & T \\ \map {\pr_2} {p, q}             & T & F & T & F \\ p \iff q                        & T & F & F & T \\ p \land q                       & T & F & F & F \\ p \uparrow q                    & F & T & T & T \\ \map \neg {p \iff q}            & F & T & T & F \\ \map {\overline {\pr_2} } {p, q} & F & T & F & T \\ \map \neg {p \implies q}        & F & T & F & F \\ \map {\overline {\pr_1} } {p, q} & F & F & T & T \\ \map \neg {p \impliedby q}      & F & F & T & F \\ p \downarrow q                  & F & F & F & T \\ \map {f_F} {p, q}               & F & F & F & F \\ \hline \end{array}$

That accounts for all $16$ of them.