Right Inverse Mapping/Examples/Real Part of Complex Number

Example of Right Inverse Mapping

Let $f: \C \to \R$ be the mapping:

$\forall z \in \C: \map f z = \map \Re z$

From Real Part as Mapping is Surjection, $f$ is a surjection.

Hence it has a right inverse $g: \R \to \C$ which, for example, can be defined as:

$\forall x \in \R: \map g x = x + i$

This right inverse is not unique.

For example, the mapping $h: \R \to \C$ defined as:

$\forall x \in \R: \map h x = x - i$

is also a right inverse.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{Q}$