Injection/Examples/2x+1 Function on Integers

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Example of Injection

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

$\forall x \in \Z: \map f x = 2 x + 1$

Then $f$ is an injection.


Let $x_1$ and $x_2$ be integers.


\(\ds \map f {x_1}\) \(=\) \(\ds \map f {x_2}\) by supposition
\(\ds \leadsto \ \ \) \(\ds 2 x_1 + 1\) \(=\) \(\ds 2 x_2 + 1\) Definition of $f$
\(\ds \leadsto \ \ \) \(\ds x_1\) \(=\) \(\ds x_2\)

Hence $f$ is an injection by definition.