# Math Help - Injective/Surjective

1. ## Injective/Surjective

I need to give examples of functions mapping from the natural numbers into the natural numbers with the following properties:

1. Surjective, not injective

2. Injective, not surjective
f(x) = 2x

3. Neither injective nor surjective

4. Bijective
f(x) = x

I think I figured out (2) and (4) but any help with the other two conditions would be helpful!

2. Hello,
Originally Posted by GoldendoodleMom
I need to give examples of functions mapping from the natural numbers into the natural numbers with the following properties:

1. Surjective, not injective
\begin{aligned} f : & \mathbb{N} & \to & \mathbb{N} \\
& n & \mapsto & f(n) \end{aligned}

$\left\{ \begin{array}{lll} f(0)=0 \\
f(1)=0 \\
f(n)=n-1 \quad \forall n \ge 2 \end{array} \right.$

2. Injective, not surjective
f(x) = 2x

3. Neither injective nor surjective
\begin{aligned} g : & \mathbb{N} & \to & \mathbb{N} \\
& n & \mapsto & g(n) \end{aligned}

$\left\{ \begin{array}{lll} g(0)=1 \\
g(1)=1 \\
g(n)=n \quad \forall n \ge 2 \end{array} \right.$

4. Bijective
f(x) = x