How to tell if $f(n) = 2 \left\lfloor\frac n2\right\rfloor$ from $\mathbb{Z}$ to $\mathbb{Z}$ is onto?

Question

How can we tell if the function $f(n) = 2 \left\lfloor\frac n2\right\rfloor$ from $\mathbb{Z}$ to $\mathbb{Z}$ is onto? Thanks!

Plug in several consecutive values of $n$ and build intuition. Then try to formalize that intuition into a proof. — user7530, Feb 11 '13 at 20:08

score 3 · Answer 1 · answered Feb 11 '13 at 20:07

A function $f:A\to B$ is onto if for every $b\in B$ there is some $a\in A$ such that $f(a)=b$. Here the function is not onto because there is no value that is mapped to $1$. Applying $f$ always yields an even integer.

How to tell if $f(n) = 2 \left\lfloor\frac n2\right\rfloor$ from $\mathbb{Z}$ to $\mathbb{Z}$ is onto?

1 Answers1