Definition:Congruence Relation

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\RR$ be an equivalence relation on $S$.


Then $\RR$ is a congruence relation for $\circ$ if and only if:

$\forall x_1, x_2, y_1, y_2 \in S: \paren {x_1 \mathrel \RR x_2} \land \paren {y_1 \mathrel \RR y_2} \implies \paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$


Also known as

Such an equivalence relation $\RR$ is also described as compatible with $\circ$.


Examples

Equal Fourth Powers over $\C$ for Multiplication

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

$\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is a congruence relation for multiplication on $\C$.


Equal Fourth Powers over $\C$ for Addition

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

$\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is not a congruence relation for addition on $\C$.


$\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ over $\Z$ for Multiplication

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

$\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then $\RR$ is not a congruence relation for multiplication on $\Z$.


$\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ over $\Z$ for Addition

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

$\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then $\RR$ is not a congruence relation for addition on $\Z$.


Also see

  • Results about congruence relations can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Congruence_Relation&oldid=567998"