Equivalence of Definitions of Injection/Definition 1 iff Definition 3

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Theorem

The following definitions of the concept of Injection are equivalent:

Definition 1

A mapping $f$ is an injection, or injective if and only if:

$\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$


That is, an injection is a mapping such that the output uniquely determines its input.

Definition 3

Let $f$ be a mapping.

Then $f$ is an injection if and only if:

$f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping

where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.


Proof

Let $f: S \to T$ be an injection by definition 1.

Thus:

$\forall x_1, x_2 \in S: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$

First we note that:

$t \in \Img f \implies \exists x \in \Dom f: \map f x = t$

thus fulfilling the condition for $f^{-1} {\restriction_{\Img f} }$ to be left-total.


Now let:

$t \in \Img f: \tuple {t, y}, \tuple {t, z} \in f^{-1}$

Thus:

\(\displaystyle \tuple {t, y}, \tuple {t, z}\) \(\in\) \(\displaystyle f^{-1} {\restriction_{\Img f} }\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tuple {y, t}, \tuple {z, t}\) \(\in\) \(\displaystyle f\) Definition of Inverse of Mapping
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map f y = t\) \(=\) \(\displaystyle \map f z\) Equality of Elements in Range of Mapping
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle z\) as $f$ is injective


So by the definition of mapping, $f^{-1} {\restriction_{\Img f} }$ is a mapping.

So $f$ is an injection by definition 3.

$\Box$


Let $f: S \to T$ be an injection by definition 3.

Then:

$f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping

where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.


We need to show that:

$\forall x, z \in \Dom f: \map f x = \map f z \implies x = z$

So, pick any $x, z \in \Dom f$ such that:

$\map f x = \map f z$

Then:

\(\displaystyle \map f x\) \(=\) \(\displaystyle \map f z\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists y \in \Dom f: \tuple {x, y}, \tuple {z, y}\) \(\in\) \(\displaystyle f\) Definition of Mapping
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tuple {y, x}, \tuple {y, z}\) \(\in\) \(\displaystyle f^{-1} {\restriction_{\Img f} }\) Definition of Inverse of Mapping
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle z\) as it is specified that $f^{-1} {\restriction_{\Img f} }$ is a mapping


So $f$ is an injection by definition 1.

$\blacksquare$


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