Surjection/Examples/Non-Surjection/2x+1 Function on Integers

Example of Mapping which is Not a Surjection

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

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

Then $f$ is not a surjection.

Proof

Consider $y = 2 n$ 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.

$\blacksquare$

Sources

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