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 with equality and the axioms for the Real numbers given in the lecture notes. Your proof should be written in the semiformal proof style using any of the shortcuts we went over.
**1. Trichotomy.**For every real number $x$ exactly one of the following is true:- $x\lt 0$
- $x=0$
- $0\lt x$

**2. Equivalence relation on the Power Set.**Define a relation $\sim$ on $\mathcal{P}(\mathbb{N})$ such that for all $U,V\in\mathcal{P}(\mathbb{N})$ $$U\sim V\text{ if and only if } \forall n\in\mathbb{N}, \left(\exists m\in U, n\lt m\right)\Leftrightarrow \left(\exists m\in V, n\lt m\right)$$ Prove $\sim$ is an equivalence relation and give a simple description of the equivalence classes.**3. Composition of bijections.**If $f\colon A\to B$ and $g\colon B\to C$ are bijections, then so is $g\circ f$, and $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$