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