Equivalence Class/Examples

Examples of Equivalence Classes

Same Age Relation

Let $P$ be the set of people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { the age of $x$ and $y$ on their last birthdays was the same}$

Then the equivalence class of $x \in P$ is:

$\eqclass x \sim = \set {\text {All people the same age as $x$ on their last birthday} }$

People Born in Same Year

Let $P$ be the set of people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ and $y$ were born in the same year}$

Then the elements of the equivalence class of $x \in P$ is:

$\eqclass x \sim = \set {\text {All people born in the same year as $x$} }$

People with Same Height

Let $A = \set {u, v, w, x, y, z}$ be a set of people.

Let $\sim$ be the relation on $A$ defined as:

$\forall \tuple {a, b} \in A \times A: x \sim y \iff \text { $a$ and $b$ are the same height}$

Let:

$u, v, w$ be the same height

$x, y, z$ be the same height but different from the height of $u, v, w$.

Then the equivalence class of $x \in A$ are $\set {u, v, w}$ and $\set {x, y, z}$.

Months that Start on the Same Day of the Week

Let $M$ be the set of months of the (calendar) year according to the (usual) Gregorian calendar.

Let $\sim$ be the relation on $M$ defined as:

$\forall x, y \in M: x \sim y \iff \text {$x$ and $y$ both start on the same day of the week}$

The set of equivalence classes under $\sim$ depends on whether the year is a leap year.

For a non-leap year, the set of equivalence classes is:

$\set {\set {\text {January}, \text {October} }, \set {\text {February}, \text {March}, \text {November} }, \set {\text {April}, \text {July} }, \set {\text {May} }, \set {\text {June} }, \set {\text {August} }, \set {\text {September}, \text {December} } }$

For a leap year, the set of equivalence classes is:

$\set {\set {\text {January}, \text {April}, \text {July} }, \set {\text {February}, \text {August} }, \set {\text {March}, \text {November} }, \set {\text {May} }, \set {\text {June} }, \set {\text {September}, \text {December} }, \set {\text {October} } }$

$z^4 = w^4$ on Complex Numbers

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 the equivalence class of $1 + i \sqrt 3$ under $\RR$ is:

$\eqclass {1 + i \sqrt 3} \RR = \set {1 + i \sqrt 3, -1 - i \sqrt 3, -\sqrt 3 + i, \sqrt 3 - i}$

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

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 the equivalence class of $1$ under $\RR$ is:

$\eqclass 1 \RR = \set {1 + 12 k: k \in \Z} \cup \set {5 + 12 k: k \in \Z}$

Congruence Modulo $\N_{< m}$

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\N_{<m}$ denote the initial segment of the natural numbers $\N$:

$\N_{<m} = \set {0, 1, \ldots, m - 1}$

Let $\RR_m$ denote the equivalence relation:

$\forall x, y \in \Z: x \mathrel {\RR_m} y \iff \exists k \in \Z: x = y + k m$

For each $a \in \N_{<m}$, let $\eqclass a m$ be the equivalence class of $a \in \N_{<m}$ under $\RR_m$ is the set:

$\eqclass a m := \set {a + z m: z \in \Z}$