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Problem 62: (Equivalence Classes Of The Mod Relation)

Consider again the relation $\mathrm{R}$ on $\mathbb{Z}$ defined by $a\mathrm{R}b$ if and only if $a\cong b \pmod{n}$. We've already shown that this relation is an equivalence relation.

  1. If we let $n=5$, then state 3 different integers that are related to $a=4$.
  2. If we let $n=3$, list the elements of the equivalence classes $ [0 ]$, $ [1] $, $ [2] $, $ [3] $, and $ [4] $.
  3. Now let $n\in\mathbb{N}$ and let $a,b\in \mathbb{Z}$. Chose integers $x$ and $y$ so that $x\cong a\pmod{n}$ and $y\cong b\pmod{n}$. Prove that $(x+y)\cong (a+b)\pmod{n}$. This proves that $ [a+b] = [a]+[b] $.

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