Congruence (Number Theory)/Integers/Examples/Modulo 2

Example of Congruence Modulo an Integer
Let $x \equiv y \pmod 2$ be defined as congruence modulo $2$:


 * $\forall x, y \in \Z: x \equiv y \pmod 2 \iff \exists k \in \Z: x - y = 2 k$

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

The equivalence classes of this equivalence relation are of the form:


 * $\left[\!\left[{x}\right]\!\right] = \left\{ {\dotsc, x - 4, x - 2, x, x + 2, x + 4, \dotsc}\right\}$

which are:

Each equivalence class has exactly one representative in the set $\left\{ {0, 1}\right\}$.