What is the number of elements of $ |\mathbb{Q} $ \ $ \mathbb{Z}| $?

Question

I know that $| \mathbb{N}|=| \mathbb{Z} | = | \mathbb{Q} |$.

So I assume after this theorem that $ |\mathbb{Q} $ \ $ \mathbb{Z}|=0 $

Is my assumption correct, and if it's correct, how can I prove it in an appropriate way?

Answer 1

$|A\backslash B|=|A|-|B|$ only if $A$ and $B$ are finite set. Here $\mathbb Q\backslash \mathbb Z$ is countable and infinite. Therefore, $$|\mathbb Q\backslash \mathbb Z|=|\mathbb N|.$$