Equivalence Relation/Examples/Non-Equivalence

Examples of Relations which are not Equivalences

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 $\Z_{>0}$ 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:

$\forall x, y \in \Z: x \mathrel \RR y \iff \text {$x + y$ is divisible by $3$}$

Then $\RR$ is not an equivalence relation.