Prove it! Math Academy | Summer Camp at Colorado State University

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 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:
  1. $x\lt 0$
  2. $x=0$
  3. $0\lt x$
- 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 $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$

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