Congruence (Number Theory)/Integers/Examples/Modulo 3

Example of Congruence Modulo an Integer
Let $x \mathrel \RR y$ be the equivalence relation defined on the integers as congruence modulo $3$:


 * $x \mathrel \RR y \iff x \equiv y \pmod 3$

defined as:


 * $\forall x, y \in \Z: x \equiv y \pmod 3 \iff \exists k \in \Z: x - y = 3 k$

That is, if their difference $x - y$ is a multiple of $3$.

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


 * $\eqclass x 3 = \set {\dotsc, x - 6, x - 3, x, x + 3, x + 6, \dotsc}$

which are:

Thus the partition of $\Z$ induced by $\RR$ is:
 * $\Bbb S = \set {\eqclass 0 3, \eqclass 1 3, \eqclass 2 3}$

Each equivalence class has exactly one representative in the set $\set {0, 1, 2}$.

So, for example, $\eqclass {17} 3 = \set {\dotsc -1, 2, 5, \dotsc, 14, 17, 20, 23, \dotsc}$