Modulo Addition is Well-Defined

Theorem
Let $$z \in \R$$.

Let $$\R_z$$ be the set of all residue classes modulo $z$ of $$\R$$.

The addition operation on $$\R_z$$, defined by the rule:


 * $$\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$$

is a well-defined operation.

This operation is called addition modulo $$z$$.

Corollary
It follows that:
 * $$\left[\!\left[{a}\right]\!\right]_z -_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a - b}\right]\!\right]_z$$

is also a well-defined operation.

Proof
We need to show that if:


 * $$\left[\!\left[{x'}\right]\!\right]_z = \left[\!\left[{x}\right]\!\right]_z$$ and
 * $$\left[\!\left[{y'}\right]\!\right]_z = \left[\!\left[{y}\right]\!\right]_z$$

then $$\left[\!\left[{x' + y'}\right]\!\right]_z = \left[\!\left[{x + y}\right]\!\right]_z$$.

Since $$\left[\!\left[{x'}\right]\!\right]_z = \left[\!\left[{x}\right]\!\right]_z$$ and $$\left[\!\left[{y'}\right]\!\right]_z = \left[\!\left[{y}\right]\!\right]_z$$, it follows from the definition of residue class modulo $z$ that $$x \equiv x' \left({\bmod\, z}\right)$$ and $$y \equiv y' \left({\bmod\, z}\right)$$.

By definition, we have:


 * $$x \equiv x' \left({\bmod\, z}\right) \implies \exists k_1 \in \Z: x = x' + k_1 z$$
 * $$y \equiv y' \left({\bmod\, z}\right) \implies \exists k_2 \in \Z: y = y' + k_2 z$$

which gives us $$x + y = x' + k_1 z + y' + k_2 z = x' + y' + \left({k_1 + k_2}\right) z$$.

As $$k_1 + k_2$$ is an integer, it follows that, by definition:
 * $$x + y \equiv \left({x' + y'}\right) \left({\bmod\, z}\right)$$.

Therefore, by the definition of residue class modulo $z$, $$\left[\!\left[{x' + y'}\right]\!\right]_z = \left[\!\left[{x + y}\right]\!\right]_z$$.

Proof of Corollary
We have:

$$ $$ $$

and as we have seen, modulo addition is well-defined for all real numbers.

Comment
Although the operation of addition modulo $$z$$ is denoted by the symbol $$+_z$$, if there is no danger of confusion, the symbol $$+$$ is often used instead.

In fact, the notation for addition of two residue classes modulo $z$ is not usually $$\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$$.

What is more normally seen is $$a + b \left({\bmod\, z}\right)$$.

Using this notation, what this result says is:

$$ $$ $$

and it can be proved in the same way.

Similarly: $$a - c \equiv b - d \left({\bmod\, z}\right)$$.

Warning
Compare this with Modulo Multiplication, which is defined only on an integer modulus.