Trivial Quotient is a Bijection

Theorem
Let $$\Delta_S$$ be the diagonal relation on a set $$S$$.

The quotient mapping $$q_{\Delta_S}: S \to S / \Delta_S$$ is a bijection.

This quotient mapping is called the trivial quotient of $$S$$.

Proof
The diagonal relation is defined as:

$$\Delta_S = \left\{{\left({x, x}\right): x \in S}\right\} \subseteq S \times S$$

From the fact that $$q_{\Delta_S}$$ is a quotient mapping, we know that it is a surjection.

It relates each $$x \in S$$ to the singleton $$\left\{{x}\right\}$$

Thus $$\left\{{x}\right\} = \left[\left[{x}\right]\right]_{\Delta_S} \subseteq S$$.

So $$q_{\Delta_S} \left({x}\right) = \left\{{x}\right\}$$, and it follows that $$\left[\left[{x}\right]\right]_{\Delta_S} = \left[\left[{y}\right]\right]_{\Delta_S} \Longrightarrow x = y$$.

Thus $$q_{\Delta_S}$$ is injective, and therefore by definition a bijection.

We may write $$q_{\Delta_S} \left({x}\right) x = x$$, which serves to emphasise its triviality.