Prove that the function $f: \mathbb{Z} \rightarrow \mathbb{N} \cup \{{0}\}$ defined by

$$f(x)= \left\{ \begin{array}{lcc} 2x & x \geq 0 \\ \\ -2x-1& x<0 \\ \\ \end{array} \right.$$ is bijective.

I don't have any familiarity with piecewise functions when dealing with bijective proofs yet. I'm assuming I have to show the bijection for two separate cases? Like for injective assume $f(x)=f(y)$ for some and arrive at $x=y$ with both cases? Would it be possible to find an inverse which would imply its bijective also?

  • 1
    $\begingroup$ Both are valid proof methods, assuming that 'two separates cases' would mean proving both injectivity and surjectivity. And yes, it's possible to explicity write a bijection for $f$. $\endgroup$ – Fimpellizieri Dec 5 '16 at 7:18

Inverse: $g:\mathbb{N}\cup\{0\}\to\mathbb{Z}$

$$g(x)= \left\{ \begin{array}{lcc} \frac{x}{2} & x \text{ is even} \\ \\ -\frac{x+1}{2}& x\text{ is odd} \\ \end{array} \right.$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.