Modulo Addition is Closed/Real Numbers

Theorem

Let $z \in \R$ be a real number.

Then addition modulo $z$ on the set of residue classes modulo $z$ is closed:

$\forall \eqclass x z, \eqclass y z \in \R_z: \eqclass x z +_z \eqclass y z \in \R_z$.

Proof

From the definition of addition modulo $z$, we have:

$\eqclass x z +_z \eqclass y z = \eqclass {x + y} z$

As $x, y \in R$, we have that $x + y \in \R$ as Real Addition is Closed.

Hence by definition of congruence, $\eqclass {x + y} z \in \R_z$.

$\blacksquare$

Also see

• Modulo Addition is Closed/Integers