Our top priority is always the safety and well-being of our students. Given the COVID-19 pandemic, we can no longer hold our in-person residential program for 2020 as originally planned. Instead, in order to do our part as we all navigate this crisis together, the Prove it! team will be offering a free online program this summer.

All spots are currently filled and we are no longer accepting additional applications. You can find information about additional summer math programs at the AMS website.

Prove the following using only the rules of natural deduction for predicate 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. Yet another DeMorgan. $\neg(\forall x,P(x)) \Leftrightarrow \exists y, \neg P(y)$
• 2. A good one to ponder. $(\exists x, P(x) \Rightarrow Q(x)) \Leftrightarrow (\forall y, P(y)) \Rightarrow (\exists z, Q(z))$
• 3. Only one Mother. $(\forall x,\exists !y,M(x,y)) \text{ and }\neg (A=B) ⇒ \neg (M(C,A) \text{ and }M(C,B))$