I have to determine if the statement is a tautology, contradiction or contingency. Been at it for days but didn't get too far.
The original question is
$$\left((\lnot p\vee z)\wedge(p\vee q)\right)\rightarrow(z\vee q)$$
I got to here
$$\left((p\wedge\lnot z)\vee(\lnot p\wedge\lnot q)\right)\vee(z\vee q)$$
Any suggestions? Not allowed to use double distribution.
Can't use truth tables, just rules of logic.
Best Answer
Starting from where you ended:
$\left((p\wedge\lnot z)\vee(\lnot p\wedge\lnot q)\right)\vee(z\vee q) \\ \lnot(\lnot p \vee z) \vee \lnot(p \vee q) \vee z \vee q \\ (\lnot(\lnot p \vee z) \vee z) \vee (\lnot(p \vee q) \vee q) \\ ((\lnot p \vee z) \rightarrow z) \vee ((p\vee q) \rightarrow q) \\ ((p \rightarrow z) \rightarrow z) \vee ((\lnot p \rightarrow q) \rightarrow q)$
We know that either $p$ or $\lnot p$ is true.
Assume $p$ is true. Then $(p\rightarrow z) \rightarrow z$ simplifies to $z \rightarrow z$, which is tautologically true.
Assume $\lnot p$ is true. Then $(\lnot p \rightarrow q) \rightarrow q$ simplifies to $q \rightarrow q$, which is tautologically true.
Thus the entire statement is equivalent to a tautology since it is always true.