# Quotient Mapping is Injection iff Equality

## Theorem

Let $\mathcal R$ be an equivalence relation on $S$.

Then the quotient mapping $q_{\mathcal R}: S \to S / \mathcal R$ is an injection if and only if $\mathcal R$ is the equality relation.

## Proof

Let $\eqclass x {\mathcal R}, \eqclass y {\mathcal R} \in S / \mathcal R$

### Sufficient Condition

Let $q_{\mathcal R}: S \to S / \mathcal R$ be an injection.

Then:

 $\displaystyle x$ $\mathcal R$ $\displaystyle y$ $\displaystyle \eqclass x {\mathcal R}$ $=$ $\displaystyle \eqclass y {\mathcal R}$ Definition of Equivalence Class $\displaystyle \leadsto \ \$ $\displaystyle \map {q_{\mathcal R} } x$ $=$ $\displaystyle \map {q_{\mathcal R} } y$ Definition of Quotient Mapping $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y$ Definition of Injection

That is:

$\mathcal R$ is the equality relation.

$\Box$

### Necessary Condition

Let $q_{\mathcal R}: S \to S / \mathcal R$ be a mapping which is specifically not an injection.

Then:

 $\, \displaystyle \exists a, b \in S, a \ne b: \,$ $\displaystyle \map {q_{\mathcal R} } a$ $=$ $\displaystyle \map {q_{\mathcal R} } b$ $q_{\mathcal R}$ is not an injection $\displaystyle \leadsto \ \$ $\displaystyle \eqclass a {\mathcal R}$ $=$ $\displaystyle \eqclass b {\mathcal R}$ Definition of Quotient Mapping $\displaystyle \leadsto \ \$ $\displaystyle a$ $\mathcal R$ $\displaystyle b$ Definition of Equivalence Class

That is:

$a \ne b$

but:

$a \mathrel {\mathcal R} b$

and so $\mathcal R$ is not the equality relation.

From Rule of Transposition it follows that:

if $\mathcal R$ is the equality relation then $q_{\mathcal R}$ is an injection.

$\blacksquare$