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:


 * $\eqclass x 1 = \Z$