The following exercise from Dr. Pinter's "A Book of Abstract Algebra" presents this exercise:

Prove that each of the following is a partition of the indicated set. Then describe the equivalence relation with that partition.

For each integer $r \in \langle0,1,2,3,4\rangle$, let $A_r$ be the set of all the integers which leave a remainder of $r$ when divided by $5$. (That is, $x \in A_r$ iff $x=5q + r$ for some integer $q$.). Prove: $\langle A_0, A_1, A_2, A_3, A_4 \rangle$ is a partition of $\mathbb{Z}$.

I wrote a number line, showing the number and its remainder modulo 5:

$$(-2 \rightarrow 3, -1 \rightarrow 4, 0 \rightarrow 0, 1 \rightarrow 1, ...)$$

But I'm not sure that's a proof - just an example. How can I prove the above?

My intuition is that modulo 5 has an equivalence relation in $\mathbb{Z}_5$, which perhaps seems obvious. But I don't know how to prove it.


In plain English:

$A_0\cup A_1\cup A_2\cup A_3\cup A_4 =\mathbf Z$:

Any integer has a remainder upon division by $5$, and this remainder is an integer between $0$ and $4$, hence any integer lies in one of the $A_i$

No $A_i$ is empty since it contains $i$.

Any two $A_i$s are disjoint since the remainder is unique (Euclid's theorem).


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