Injective/Surjective

• Sep 10th 2008, 08:14 AM
GoldendoodleMom
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!
• Sep 10th 2008, 08:19 AM
Moo
Hello,
Quote:

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

\displaystyle \begin{aligned} f : & \mathbb{N} & \to & \mathbb{N} \\ & n & \mapsto & f(n) \end{aligned}

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

Quote:

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

Quote:

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

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

Quote:

4. Bijective
f(x) = x
(Clapping)