I agree. The notations and probably refer to Peirce's arrow (NOR) and Sheffer stroke (NAND), respectively.
This depends on the assignment. There are many ways of proving a propositional formula, from constructing a truth table to giving a derivation in one of many formal systems.
Sorry, I also had to prove the formula ¬(p ↑ q)⇐⇒(¬p ↓¬q) so I added that information to the columns. since the columns for ¬(p ↓ q) equal the columns for (¬p ↑¬q) that formula is true. Since the columns for ¬(p ↑ q) equal the columns for (¬p ↓¬q) that formula is true.
Thank you for challanging me to finish thinking this problem through. I have suffered a recent brain injury and my thought process suffers from this.