Congruence (Number Theory)/Integers/Examples/Modulo 1

Example of Congruence Modulo an Integer

Let $x \equiv y \pmod 1$ be defined on the integers as congruence modulo $1$:

$\forall x, y \in \Z: x \equiv y \pmod 1 \iff \exists k \in \Z: x - y = k$

That is, if their difference $x - y$ is an integer.

The equivalence classes of this equivalence relation is the set of integers:

$\eqclass x 1 = \Z$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.5$. Congruence of integers: Example $38$