Injection/Examples/2x Function on Integers

Example of Injection which is Not a Surjection

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

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

Then $f$ is an injection, but not a surjection.

Let $x_1$ and $x_2$ be integers.

Then:

$\ds \map f {x_1}$

$=$

$\ds \map f {x_2}$

by supposition

$\ds \leadsto \ \ $

$\ds 2 x_1$

$=$

$\ds 2 x_2$

Definition of $f$

$\ds \leadsto \ \ $

$\ds x_1$

$=$

$\ds x_2$

Hence $f$ is an injection by definition.

$\Box$

Now consider $y = 2 n + 1$ for some $n \in \Z$.

There exists no $x \in \Z$ such that $\map f x = y$.

Thus by definition $f$ is not a surjection.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $8$
1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.4$: Functions: Problem Set $\text{A}.4$: $23$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $5$