prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality

Question

I have to prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality. I think I have bijective function($f(\langle a,b\rangle)=a-b$) between thse sets, but I don't know how to prove that this function is bijective. I know I have to prove that $f$ is injection and surjection.

Thanks.

Answer 1

You are on the right path, note that if $(a,b)\neq (c,d)$ then either $a\neq c$ in which case $a-1\leq a-b<a$ and $c-1\leq c-d<c$ so we must have $a-b\neq c-d$; and if $a=c$ then $b\neq d$ and similarly it follows.

To show surjectivity suppose that $q\in\Bbb Q$, find some $a\in\Bbb Z$ and $b\in\Bbb Q\cap(0,1]$ such that $a-b=q$. I leave that to you.