Definition:Union Mapping

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Definition

Let:

$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$
$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$

Let $f_1$ and $f_2$ be combinable, that is, that they agree on $S_1 \cap S_2$.


Then the union mapping $f = f_1 \cup f_2$ of $f_1$ and $f_2$ is:

$f: S_1 \cup S_2 \to T_1 \cup T_2: \map f s = \begin{cases} \map {f_1} s : & s \in S_1 \\ \map {f_2} s : & s \in S_2 \end{cases}$


Finite Set of Mappings

Let $S = \set {f_1, f_2, \ldots, f_n}$ denote a finite set of mappings.


The union mapping $f$ of $S$ is defined when:

$\forall i, j \in \set {1, 2, \ldots, n}: f_i$ and $f_j$ are combinable

and is defined as:

$\forall x \in \displaystyle \bigcup \set {\Dom {f_i}: i \in \set {1, 2, \ldots, n} } x \in \Dom {f_i} \implies f = \map {f_i} x$


Family of Mappings

Let $I$ be an indexing set.

Let $F = \family {f_i}_{i \mathop \in I}$ be a family of mappings indexed by $I$

The union mapping $f$ of $F$ is defined when:

$\forall i, j \in I: f_i$ and $f_j$ are combinable

and is defined as:

$\forall x \in \displaystyle \bigcup \set {\Dom {f_i}: i \in I} x \in \Dom {f_i} \implies f = \map {f_i} x$


Also known as

The union mapping (of $f_1$ and $f_2$) can also be seen referred to as the combined mapping (of $f_1$ and $f_2$).


Examples

Absolute Value Function

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 $x$.


Also see

  • Results about union mappings can be found here.


Sources