Mapping/Mistakes/Mapping not Well-Defined

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Example of Mistake in Definition of Mapping

This example of an attempted definition of a mapping contains a mistake.

$\theta: \Q \to \Z$ defined as: $\forall m, n \in \Z, n \ne 0: \dfrac m n \mapsto m + n$


Explanation

The mapping $\theta$ is not well-defined.

Let $x \in \Q$ be such that $x = \dfrac 1 2$.

Then:

$\theta \paren {\dfrac 1 2} = 1 + 2 = 3$
$\theta \paren {\dfrac 2 4} = 2 + 4 = 6$
$\theta \paren {\dfrac 3 6} = 3 + 6 = 9$

and so on.

Thus $\theta \paren {\dfrac 1 2} = \set {3, 6, 9, \ldots}$


If $\theta$ were to be defined as:

$\theta: \Z \times \Z: \tuple {m, n} \mapsto m + n$

that would be valid.

However, the fact that each of $\dfrac 1 2, \dfrac 2 4, \dfrac 3 6, \ldots$ all refer to the same $x \in \Q$ renders $\theta$ as not actually a mapping.

$\blacksquare$


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