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

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Example of Congruence Modulo an Integer

Let $x \equiv y \pmod 2$ be defined on the integers 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:

$\eqclass x 2 = \set {\dotsc, x - 4, x - 2, x, x + 2, x + 4, \dotsc}$

which are:

\(\ds \eqclass 0 2\) \(=\) \(\ds \set {\dotsc, -4, -2, 0, 2, 4, \dotsc}\) that is, the even integers
\(\ds \eqclass 1 2\) \(=\) \(\ds \set {\dotsc, -3, -1, 1, 3, 5, \dotsc}\) that is, the odd integers


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


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