Congruence (Number Theory)/Integers/Examples/Modulo 2
Jump to navigation
Jump to search
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 $\set {0, 1}$.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $5 \ \text{(i)}$
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Ponderable $2.3.4$