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 modulo 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.

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:

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

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 $z$:

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

$\blacksquare$

Proof of Corollary

We have:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left[\!\left[{a}\right]\!\right]_z -_z \left[\!\left[{b}\right]\!\right]_z\)	\(=\)	\(\displaystyle \left[\!\left[{a - b}\right]\!\right]_z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{a + \left({-b}\right)}\right]\!\right]_z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{-b}\right]\!\right]_z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

$\blacksquare$

Warning

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

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.6$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 11$: Example $11.2$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.6$: Theorem $4$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $2$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 19 \beta$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 18.4$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Proposition $2.31$