Congruence (Number Theory)/Examples/Modulo 1

Example of Congruence Modulo an Integer

Let $x \equiv y \pmod 1$ be defined as congruence on the real numbers modulo $1$:

$\forall x, y \in \R: x \equiv y \pmod 1 \iff \exists k \in \Z: x - y = k$

That is, if their difference $x - y$ is an integer.

The equivalence classes of this equivalence relation are of the form:

$\eqclass x 1 = \set {\dotsc, x - 2, x - 1, x, x + 1, x + 2, \dotsc}$

Each equivalence class has exactly one representative in the half-open real interval:

$\hointr 0 1 = \set {x \in \R: 0 \le x < 1}$

Also see

• Definition:Modulo 1

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $1 \ \text {(i)}$
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Example $2.3.6$