Injection/Examples/2x+1 Function on Integers

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.

Proof

Let $x_1$ and $x_2$ be integers.

Then:

\(\ds \map f {x_1}\)

\(=\)

\(\ds \map f {x_2}\)

by supposition

\(\ds \leadsto \ \ \)

\(\ds 2 x_1 + 1\)

\(=\)

\(\ds 2 x_2 + 1\)

Definition of $f$

\(\ds \leadsto \ \ \)

\(\ds x_1\)

\(=\)

\(\ds x_2\)

Hence $f$ is an injection by definition.

$\Box$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $2 \ \text {(iv) (a)}$