Describing Equivalence Classes using set builder notation

Question

How would you describe all the equivalence classes for the relation: $congruence$ $modulo$ $5$ over $Z$, using set builder notation?

Answer 1

Note that the equivalence classes must be subsets of $\Bbb Z$. Using set-builder notation, the equivalence classes are $$S_a=\{\,a+5k\mid k\in{\Bbb Z}\,\}$$ for $a=0,1,2,3,4$.

Answer 2

The equivalence classes of $\mathbb{Z}_5$ will be given by it's elements.

Here: $\mathbb{Z}_5=\{0,1,2,3,4\}$

Thus, you can read off the equivalence classes of $\mathbb{Z}_5$.

$\bf{Additional}$: Yes, for each class to obtain elements in that specific class you begin with the element which names the class and add any multiple of your modulus.

Answer 3

These will be expressions for the set of equivalence classes; each equivalence class will be an element of the set.

Using set builder notation twice: $\big\{\ \{n + 5m: m \in \mathbb{Z} \} : n \in \mathbb{Z} \ \big\}$, or more economically, $\big\{\ \{n + 5m: m \in \mathbb{Z} \} : n = 0,1,2,3,4 \ \big\}$

Using it just once: $\{ n + 5\mathbb{Z} : n \in \mathbb{Z} \}$, or more economically, $\{n + 5\mathbb{Z} : n = 0,1,2,3,4 \}$.

Most economically (and definitionally): $\mathbb{Z}/5\mathbb{Z}$.