Left Inverse Mapping/Examples/Inclusion of Reals in Complex Plane
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Example of Right Inverse Mapping
Let $i_\R: \R \to \C$ be the inclusion mapping of the real numbers into the complex plane:
- $\forall x \in \R: \map {i_\R} z = x + 0 i$
From Inclusion Mapping is Injection, $i_\R$ is an injection.
Hence it has a left inverse $g: \C \to \R$ which, for example, can be defined as:
- $\forall z \in \C: \map g z = \map \Re z$
This left inverse is not unique.
For example, the mapping $h: \R \to \C$ defined as:
- $\forall z \in \C: \map g z = \map \Re z + \map \Im z$
is also a left inverse.
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{Q}$