# Proving Expression is Contingency with Logical Equivalences

I am trying to prove that $$\lnot[(q \vee r) \wedge ((p \vee q) \wedge (\lnot p \vee r))]$$ is a contingency using logical equivalence rules. I have tried various steps and keep getting stuck in loops, or making wrong moves and getting tautology or contradiction.

The last steps I tried were dropping the brackets surrounding all the or's (commutativity) and getting $$\lnot[q \vee r \vee p \vee q \wedge (\lnot p\vee r)]$$ then followed by:

$$\lnot[q \vee p \vee r \wedge (\lnot p\vee r)] \mbox{ (idempotent)}$$

$$\lnot[q \vee p \vee r \wedge (\lnot p \vee r)] \mbox{ (commutativity again)}$$

$$\lnot(q \vee p \vee r) \mbox{ (absorption)}$$

For one I feel like I'm making some either useless, or incorrect steps here, as well I don't see how $$\lnot(q \vee p \vee r)$$ would prove it's a contingency. I've tried looking at examples and other questions related to this topic, but I just can't seem to find anything that makes sense given the question. Any help would be greatly appreciated, thank you!

Now, it doesn't take much experience with logic to recognize $$\neg (q \lor p \lor r)$$ as a contingency, so for some audiences what you did will be enough. However, for a hard proof, you probably want to come up with two different truth-valuie assignment: one that sets the statement to True, and another on that sets the statement to false.
For example, to set the statement to false, we can set $$p=q=r=True$$, for then the statement evaluates to $$\neg (q \lor p \lor r) =\neg (T \lor T \lor T) = \neg T = F$$