Union Mapping/Examples/Absolute Value

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Examples of Union Mappings

Let $f_1: \R_{\ge 0} \to \R$ be the real function defined on the set of positive real numbers $\R_{\ge 0}$ as:

$\forall x \in \R_{\ge 0}: \map {f_1} x = x$

Let $f_2: \R_{\le 0} \to \R$ be the real function defined on the set of negative real numbers $\R_{\le 0}$ as:

$\forall x \in \R_{\le 0}: \map {f_2} x = -x$


Then:

$f_1$ and $f_2$ are combinable mappings

and:

the union mapping $f = f_1 \cup f_2$ is:
$\forall x \in \R: \map f x = \size x$
where $\size x$ denotes the absolute value of $x$.


Proof

We have that:

$\Dom {f_1} \cap \Dom {f_2} = \set 0$

and:

$\map {f_1} 0 = 0 = \map {f_2} 0$

Hence by definition $f_1$ and $f_2$ are combinable.

Then we have that:

$\forall x \in \R: \map f x = \begin {cases} x & : x \ge 0 \\ -x & : x \le 0 \end {cases}$

and the result follows by definition of the absolute value of $x$.

$\blacksquare$


Sources