Quotient Theorem for Sets/Examples/Real Square Function

Example of Use of Quotient Theorem for Sets
Let $f: \R \to \R$ denote the square function:
 * $\forall x \in \R: \map f x = x^2$

We define $\RR_f \subseteq S \times S$ to be the relation:


 * $\tuple {x_1, x_2} \in \RR_f \iff {x_1}^2 = {x_2}^2$

that is:


 * $x_1 \mathrel {\RR_f} x_2 \iff x_1 = \pm x_2$

The quotient set of $\R$ induced by $\RR_f$ is thus the set $\R / \RR_f$ of $\RR$-classes of $\RR$:

Hence the quotient mapping $q_{\RR_f}$:
 * $q_{\RR_f}: \R \to \R / \RR_f: \map {q_{\RR_f} } x = \eqclass x {\RR_f}$

puts $x$ into its equivalence class $\set {x, -x}$.

We note in passing that $\eqclass x {\RR_f}$ has $2$ elements unless $x = 0$.

The renaming mapping is defined as:


 * $r: \R / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = x^2$

where $\Img f = \R_{\ge 0}$.

Finally the inclusion mapping is defined as:
 * $i_{\R_{\ge 0} }: \R_{\ge 0} \to \R: \map {i_{\R_{\ge 0} } } x = x$