Quotient Theorem for Sets/Examples
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Examples of Use of Quotient Theorem for Sets
Real Square Function
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 $\RR$:
\(\ds \R / \RR_f\) | \(:=\) | \(\ds \set {\eqclass x {\RR_f}: x \in \R}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\set {x, -x}: x \in \R}\) |
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:
- $i_{\R_{\ge 0} }: \R_{\ge 0} \to \R: \map {i_{\R_{\ge 0} } } x = x$