Definition:Union Mapping
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:
- $\forall x \in \ds \bigcup \set {\Dom {f_i}: i \in \set {1, 2, \ldots, n} } x \in \Dom {f_i} \implies f = \map {f_i} x$
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:
- $\forall x \in \ds \bigcup \set {\Dom {f_i}: i \in I} x \in \Dom {f_i} \implies f = \map {f_i} x$
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
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Restrictions and Extensions