Binary Truth Functions

Theorem
There are $16$ distinct binary truth functions:


 * Two constant functions:
 * $f_F \left({p, q}\right) = F$
 * $f_T \left({p, q}\right) = T$


 * Two projections:
 * $\operatorname{pr}_1 \left({p, q}\right) = p$
 * $\operatorname{pr}_2 \left({p, q}\right) = q$


 * Two negated projections:
 * $\overline {\operatorname{pr}_1} \left({p, q}\right) = \neg p$
 * $\overline {\operatorname{pr}_2} \left({p, q}\right) = \neg q$


 * 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: $\neg \left({p \iff q}\right)$


 * Two negated conditionals:
 * $\neg \left({p \implies q}\right)$
 * $\neg \left({q \implies p}\right)$


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

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

These can be depicted in a truth table as follows:

$\begin{array}{|r|cccc|} \hline p                                                  & T & T & F & F \\ q                                                  & T & F & T & F \\ \hline f_T \left({p, q}\right)                            & T & T & T & T \\ p \lor q                                           & T & T & T & F \\ p \impliedby q                                     & T & T & F & T \\ \operatorname{pr}_1 \left({p, q}\right)            & T & T & F & F \\ p \implies q                                       & T & F & T & T \\ \operatorname{pr}_2 \left({p, q}\right)            & T & F & T & F \\ p \iff q                                           & T & F & F & T \\ p \land q                                          & T & F & F & F \\ p \uparrow q                                       & F & T & T & T \\ \neg \left({p \iff q}\right)                       & F & T & T & F \\ \overline {\operatorname{pr}_2} \left({p, q}\right) & F & T & F & T \\ \neg \left({p \implies q}\right)                   & F & T & F & F \\ \overline {\operatorname{pr}_1} \left({p, q}\right) & F & F & T & T \\ \neg \left({p \impliedby q}\right)                 & F & F & T & F \\ p \downarrow q                                     & F & F & F & T \\ f_F \left({p, q}\right)                            & F & F & F & F \\ \hline \end{array}$

That accounts for all sixteen of them.