4.I.2D

(a) Let $\sim$ be an equivalence relation on a set $X$. What is an equivalence class of $\sim ?$ Prove that the equivalence classes of $\sim$ form a partition of $X$.

(b) Let $\mathbb{Z}^{+}$be the set of all positive integers. Let a relation $\sim$ be defined on $\mathbb{Z}^{+}$by setting $m \sim n$ if and only if $m / n=2^{k}$ for some (not necessarily positive) integer $k$. Prove that $\sim$ is an equivalence relation, and give an example of a set $A \subset \mathbb{Z}^{+}$that contains precisely one element of each equivalence class.

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