Real Part as Mapping is Surjection
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Theorem
Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as:
- $\forall z \in \C: \map f z = \map \Re z$
where $\map \Re z$ denotes the real part of $z$.
Then $f$ is a surjection.
Proof
Let $x \in \R$ be a real number.
Let $y \in \R$ be an arbitrary real number.
Let $z \in \C$ be the complex number defined as:
- $z = x + i y$
Then we have:
- $\map \Re z = x$
That is:
- $\exists z \in \C: \map f z = x$
The result follows by definition of surjection.
$\blacksquare$
Also see
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products