I am having trouble with this problem:
Give an example of a function $f:Z \rightarrow N$ that is
a. surjective but not injective
b. injective but not surjective
Work: I came up with examples such as $f=2|x-1|$ only to realize that it is not injective or surjective.
Best Answer
Bijection $\mathbb{Z} \to \mathbb{N}$:
$$f(x) = \left|2x-\frac{1}{2}\right|+\frac{1}{2}$$
Injections $\mathbb{Z} \to \mathbb{N}$:
$$g(x) = f(2x)\quad \text{ or } \quad g'(x) = 2f(x)$$
Surjections $\mathbb{Z} \to \mathbb{N}$:
$$h(x) = f\left(\left\lfloor\frac{x}{2}\right\rfloor\right) \quad \text{ or } \quad h'(x) = \left\lfloor\frac{f(x)}{2}\right\rfloor$$
I hope this helps $\ddot\smile$