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