Equivalence Relation/Examples
Examples of Equivalence Relations
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}$
That is, that $x$ and $y$ are the same age.
Then $\sim$ is an equivalence relation.
Same Parents 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 { both of the parents of $x$ and $y$ are the same}$
Then $\sim$ is an equivalence relation.
People with Same First Name
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 $y$ have the same first name}$
Then $\sim$ is an equivalence relation.
Books with Same Number of Pages
Let $P$ be the set of books.
Let $\sim$ be the relation on $P$ defined as:
- $\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ and $y$ have the same number of pages}$
Then $\sim$ is an equivalence relation.
Even Sum Relation
Let $\Z$ denote the set of integers.
Let $\RR$ denote the relation on $\Z$ defined as:
- $\forall x, y \in \Z: x \mathrel \RR y \iff x + y \text { is even}$
Then $\RR$ is an equivalence relation.
The equivalence classes are:
- $\eqclass 0 \RR$
- $\eqclass 1 \RR$
Months that Start on the Same Day of the Week
Let $M$ be the set of months of the 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}$
Then $\sim$ is an equivalence relation.
Points on Same Horizontal Line
Equivalence Relation/Examples/Points on Same Horizontal Line
$z^4 = w^4$ on Complex Numbers
Let $\C$ denote the set of complex numbers.
Let $\RR$ denote the relation on $\C$ defined as:
- $\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$
Then $\RR$ is an equivalence relation.
$\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ on Integers
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 $\RR$ is an equivalence relation.
Examples of Non-Equivalence Relations
Different 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 not the same}$
Then $\sim$ is not an equivalence relation.
Is the Mother Of is not Equivalence
Let $P$ denote 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 { $x$ is the mother of $y$}$
Then $\sim$ is not an equivalence relation.
Is the Sister Of is not Equivalence
Let $P$ denote 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 { $x$ is the sister of $y$}$
Then $\sim$ is not an equivalence relation.
Common Ancestor 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 { $x$ and $y$ have an ancestor in common}$
Then $\sim$ is not an equivalence relation.
Greater Than is not Equivalence
Let $\R$ denote the set of real number.
Let $>$ denote the usual relation on $\R$ defined as:
- $\forall \tuple {x, y} \in \R \times \R: x > y \iff \text { $x$ is (strictly) greater than $y$}$
Then $>$ is not an equivalence relation.
$\forall x, y \in \R: x + y \in \Z$ is not Equivalence
Let $\R$ denote the set of real numbers.
Let $\sim$ denote the relation defined on $\R$ as:
- $\forall \tuple {x, y} \in \R \times \R: x \sim y \iff x + y \in \Z$
Then $\sim$ is not an equivalence relation.
Divisor Relation is not Equivalence
Let $\Z$ denote the set of (strictly) positive integers.
Let $x \divides y$ denote that $x$ is a divisor of $y$
Then $\divides$ is not an equivalence relation.
Sum of Integers is Divisible by $3$ is not Equivalence
Let $\Z$ denote the set of integers.
Let $\RR$ denote the relation on $\Z$ defined as:
- $\forall x, y \in \Z: x \mathrel \RR y \iff x + y \text { is divisible by } 3$
Then $\RR$ is not an equivalence relation.