How can i get a tautology truth table from using 3 variables?

Question

I am looking to use the variables p, q and r to create a truth table which concludes to a tautology.

Answer 1

$(p \vee \lnot p) \wedge (q \vee \lnot q) \wedge (r \vee \lnot r)$.

Answer 2

There are lots and lots of examples:

• start with obvious: $p \lor q \lor r \lor \mathtt{true}$,
• or alsmost obvious: $p \lor q \lor r \lor \neg p$,
• something with conjunction: $T_1(p) \land T_2(q) \land T_3(r)$ for any tautologies $T_1,T_2,T_3$ on one variable like $x \lor \neg x$,
• something yet more complicated $T_1(p,q) \land T_2(q,r) \land T_3(r,p)$ for any tautologies $T_1,T_2,T_3$ on two variables like $(x \to y) \lor (y \to x)$,
• something that does depend on three variables in a non-trivial way $$(p \leftrightarrow q) \lor (q \leftrightarrow r) \lor (r \leftrightarrow p),$$ that one is true in classical logic, because we have three variables while only two possible values, so one of the values has to be shared.

I hope this helps $\ddot\smile$