Modulo Addition is Closed/Real Numbers
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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, $\eqclass {x + y} z \in \R_z$.
$\blacksquare$