Preimages All Exist iff Surjection/Proof 1

Theorem

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

Let $f^{-1}$ be the inverse of $f$.

Let $\map {f^{-1} } t$ be the preimage of $t \in T$.

Then $\map {f^{-1} } t$ is empty for no $t \in T$ if and only if $f$ is a surjection.

Proof

Necessary Condition

We use a Proof by Contraposition.

To that end, suppose:

$\exists t \in T: \map {f^{-1} } t = \O$

That is:

$\neg \paren {\forall t \in T: \exists s \in S: \map f s = t}$

So, by definition, $f: S \to T$ is not a surjection.

From Rule of Transposition it follows that if $f$ is a surjection: $\neg \exists t \in T: \map {f^{-1} } t = \O$

$\Box$

Sufficient Condition

We again use a Proof by Contraposition.

To that end, suppose $f$ is not a surjection.

Then by definition:

$\exists t \in T: \neg \paren {\exists s \in S: \map f s = t}$

That is:

$\exists t \in T: \map {f^{-1} } t = \O$

From Rule of Transposition it follows that if $\neg \exists t \in T: \map {f^{-1} } t = \O$, then $f$ is a surjection.

$\blacksquare$