Determine whether the following expression is a tautology, a contingency, or a contradiction by using the logical equivalences As a practice problem, I'm asked to determine whether the following proposition is a tautology, contradiction, or contingency through the use of logical equivalences.
I get how to determine what truth value the proposition is and I understand most of the logic rules, but I'm stumped on what the first step would be on solving this one. I can't use distribution on ( (  ∨  ) ∧ ( ¬  ∨  ) ) because of the negation symbol on p. If I use de morgan's by moving that outer negation inside, I don't see how it would make a difference because all it'd do is just flip the signs. If I used association and moved the bracket to ( (  ∨  ) ∨ (  ∨  )) I still can't use distribution because all the symbols are a ∨.
¬ ( (  ∨  ) ∨ ( (  ∨  ) ∧ ( ¬  ∨  ) ) )
Any help or push in the right direction would be appreciated.
 A: I will in the following show the reduction of $\neg ((q \lor r) \lor ((p \lor q) \land (\neg p \lor r)))$:
By De Morgan's laws:
$\equiv \neg (q \lor r) \land \neg ((p \lor q) \land (\neg p \lor r))$
By De Morgan's laws:
$\equiv \neg (q \lor r) \land \neg (p \lor q) \lor \neg (\neg p \lor r)$
By De Morgan's laws:
$\equiv (\neg q \land \neg r) \land (\neg p \land \neg q) \lor (p \land \neg r)$
$\equiv (\neg q \land \neg r \land \neg p) \lor (p \land \neg r)$
$\equiv (\neg q \lor p) \land (\neg r \lor p) \land (\neg p \lor p) \land (\neg q \lor \neg r) \land (\neg r \lor \neg r) \land (\neg p \lor \neg r)$
$\equiv (\neg q \lor p) \land (\neg r \lor p) \land T \land (\neg q \lor \neg r) \land T \land (\neg p \lor \neg r)$
$\equiv (\neg q \lor p) \land (\neg r \lor p) \land (\neg q \lor \neg r) \land (\neg p \lor \neg r)$
$\equiv (\neg q \lor p) \land (\neg q \lor \neg r) \land (\neg r \lor (p \land \neg q))$
$\equiv (\neg q \lor (p \land \neg r)) \land (\neg r \lor (p \land \neg q))$
As you can see, I have used De Morgan's laws or the first 3 steps. Even though it doesn't seem like it will make any difference swapping the negation sign and inverting the boolean operator, it might actually give a surprising result since some parentheses may be eliminated because they contain a tautology or contradiction. This is exactly what I have done after the first 3 steps.
