Relations & modular artithmetic

Question

Given the following partition on the set N:{ n being natural : n = 7k+p} , where p= 0,1,2,3,4,5,6.

1) Find an equivalence relation ~ on the set N that partitions N into the sets mentioned in the partitions above

Answer : mRn iff m ≅ n(mod 7)

I didn't recognize how the answer is obtained

Answer 1

The sets $$\{n \in \mathbb{N}: n=7k+p \text{ for some } k \in \mathbb{Z} \}$$ for $p \in \{0,1,\ldots,6\}$ are the equivalence classes (congruence classes) under the equivalence relation defined by $$n R m \quad\text{ if and only if }\quad m \equiv n \pmod 7.$$

We could derive formally this as follows; the equivalence class containing $a \in \mathbb{N}$ is: \begin{align*} [a] &= \{n \in \mathbb{N}:n \equiv a \pmod 7\} \\ &= \{n \in \mathbb{N}:7 \text{ divides } n-a\} \\ &= \{n \in \mathbb{N}:n-a=7r \text{ for some } r \in \mathbb{Z}\} \\ &= \{n \in \mathbb{N}:n=7r+a \text{ for some } r \in \mathbb{Z}\} \\ &= \{n \in \mathbb{N}:n=7k+(a \text{ mod } 7) \text{ for some } k \in \mathbb{Z}\} \end{align*} where $(a \text{ mod } 7)$ denotes the least non-negative residue of $a$ modulo $7$. The last step is just a change of variables.

The possible values of $(a \text{ mod } 7)$ belong to $\{0,1,\ldots,6\}$.

Answer 2

Hint $\ $ By definition, the equivalence class of $\,n\,$ is $\ (n\ {\rm mod}\ 7) + 7\,\Bbb Z\, =\, n + 7\,\Bbb Z,\,$ and

$$m+7\,\Bbb Z = n+7\,\Bbb Z\,\iff\, m\equiv n\!\!\pmod 7\qquad $$