### The Prove it! Math Academy residential summer program will not be offered in 2023. Big changes are in the works to provide our unique curriculum and high-quality resources to a wider audience of math enthusiasts in 2023. Check this site for updates and announcements.

Prove the following using only the rules of natural deduction for propositional logic given in the lecture notes. Your proof should be written in the formal proof style. You can use Lurch to check your proofs. If you finish these and want more practice there are more problems in the lecture notes you can try. Feel free to ask an instructor for help if you get stuck.

• 1. The Most Beautiful Tautology? $P \Rightarrow (P \Leftrightarrow P \text{ or } (\neg P \text{ and }P))$Bonus: Do you see why is it beautiful?
• 2. Alternate form of implication. $(P \Rightarrow Q) \Leftrightarrow (\neg P \text{ or } Q)$
• 3. Associativity of or. $P \text{ or } (Q \text{ or } R) \Leftrightarrow (P \text{ or } Q) \text{ or } R$
• 4. Pierce’s Law $((P \Rightarrow Q) \Rightarrow P) \Rightarrow P$