### The Prove it! Math Academy summer program will not be offered in 2021. Big changes are in the works to provide our unique curriculum and high-quality resources to a wider audience of math enthusiasts in 2022. Check this site for updates or follow us on Facebook for announcements.

- 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 $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$