Is this a correct mathematical statement about integers mod n and congruence mod n?

Question

Let's say we have the congruence: $b \equiv a \mod 20$

Is it then correct to say that $(b \equiv a \mod 20 )= \mathbb{Z}_{20}$ where $\mathbb{Z}_{20}$ is the set of residue classes or set of equivalence classes of the congruence $b \equiv a \mod 20?$ (So $\mathbb{Z}_{20}=\{{[0],[1],[2],[3],[4], ..., [19]}$}

Answer 1

No, that is not proper notation. It seems you wish to state that all possible remainders yields a complete set of representatives for $\,\Bbb Z_{20}.\,$ This can be stated (close to your presentation) as follows:

$$\{\, b\in \Bbb Z\ :\ b = (a\bmod 20)\ \ \text{for some } a \in \Bbb Z\} = \Bbb Z_{20}$$

Equivalently, $\ \Bbb Z\bmod 20 = \Bbb Z_{20}\,$ if you define $\bmod$ on sets (but some may object to such notation).

If you wish to work with equivalence (congruence) classes vs. remainders then it is simpler

$$\{ \,[\,a\,]_{20}\ :\ a\in \Bbb Z\} = \Bbb Z_{20} = \Bbb Z/20\Bbb Z$$

Answer 2

Saying "$(b \equiv a \mod 20 )= \mathbb{Z}_{20}$" is not correct at all. At best, the left-hand side would be interpreted as an equivalence class of integers (though not in the way you've written it), which is an element of $\mathbb{Z}_{20}$. This would be like taking the true statement $a \in A$ and instead writing $a=A$. There's a world of difference between $\in$ and $=$. Maybe this was a typo on your part.

If you you wanted to say that the class of $a$ is a member of $\mathbb{Z}_{20}$, acceptable standard notations would be $[a] \in \mathbb{Z}_{20}$ or $\overline{a} \in \mathbb{Z}_{20}$.

In short, if $A$ has a relation $\equiv$ on it, you may say that elements $a, b \in A$ are related, written $a \equiv b$. This is a statement that is either true or false. You may form equivalence classes, which are subsets of $A$: this would be something like $[a] = \{b \in A \mid a \equiv b\}$. You may also form the set of these equivalence classes, in which $[a]$ is an element (not a subset) and $a$ is neither element nor subset.

Answer 3

Note that “$b\equiv a\pmod{20}$” is a proposition, that can be either true or false, depending on the value of $a$ and $b$. To the contrary, “$\mathbb{Z}_{20}$” is a set.

The two objects live in different realms, so they cannot be “equal”; there is a link between the two, because congruence modulo $20$ can be dealt with computations in $\mathbb{Z}_{20}$.