Injection/Examples/Negative Function on Integers
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Example of Mapping which is Not an Injection
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = -x$
Then $f$ is an injection.
Proof
Let $x_1$ and $x_2$ be integers.
Then:
\(\ds \map f {x_1}\) | \(=\) | \(\ds \map f {x_2}\) | by supposition | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -x_1\) | \(=\) | \(\ds -x_2\) | Definition of $f$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_1\) | \(=\) | \(\ds x_2\) |
Hence $f$ is an injection by definition.
$\Box$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $2 \ \text {(ii) (a)}$