Injection has Surjective Left Inverse Mapping/Proof 2

Theorem

Let $S$ and $T$ be sets such that $S \ne \O$.

Let $f: S \to T$ be a injection.


Then there exists a surjection $g: T \to S$ such that:

$g \circ f = I_S$


Proof

By Injection iff Left Inverse, $f$ has a left inverse $g: T \to S$.

By Left Inverse Mapping is Surjection, $g$ is a surjection.

$\blacksquare$


Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Injection iff Left Inverse.

This is one of the logical axioms that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.

This in turn invalidates this theorem from an intuitionistic perspective.

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