Definition:Congruence (Number Theory)/Modulo Zero

Definition
Let $x, y \in \R$.

The relation congruence modulo zero is defined as:
 * $x \equiv y \pmod 0 \iff x \bmod 0 = y \bmod 0 \iff x = y$

and:
 * $x \equiv y \pmod 0 \iff \exists k \in \Z: x - y = 0 \cdot k = 0 \iff x = y$

This definition is consistent with the general definition of congruence modulo $z$ for any $z \in \R$.

Also see

 * Equivalence of Definitions of Congruence


 * Congruence Modulo $m$ is Equivalence Relation