Quotient Mapping/Examples/Congruence Modulo 3

Example of Quotient Mapping
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 $3$.

From Congruence Modulo $3$, the quotient set induced by $\RR$ is:
 * $\Z / \RR = \set {\eqclass 0 3, \eqclass 1 3, \eqclass 2 3}$

Hence the quotient mapping $q_\RR: \Z \to \Z / \RR$ is defined as:
 * $\forall x \in \Z: \map {q_\RR} x = \eqclass x 3 = \set {x + 3 k: k \in \Z}$