I'm reviewing discrete math a second time (after it being over a decade since I took the course in college).
How does one go from this step: $(\neg p \lor \neg q) \lor (p \lor q)$ to this one: $(\neg p \lor p) \lor (\neg q \lor q)$?
Rowsen's text just says "by the associative and commutative laws for disjunction".
What exactly are the associative step(s) involved in this transition? I understand the simple concept of commutative laws. Basically, I'm looking for a more rigorous explanation of how to arrive at this step than the one Rowsen supplies.
Also, regarding commutative, I would think that that the parentheses don't matter since all of the variables are connected by disjunctions only. Thus couldn't one also reason that $(\neg p \lor \neg q) \lor (p \lor q) \equiv \neg p \lor \neg q \lor p \lor q$? If so than it would seem easy to rearrange this by the commutative law. Then just add back the parentheses to get the desired result (in this case the goal being to arrive at $T \lor T$ by negation).