Which of the following statements is or are always true, with $m$ and $n$ $∈$ $\mathbb R$?
I. If $m$ is a rational no negative number and $n$ an irrational number, then $m n$ is irrational.
II. If $m$ and $n$ are rationals, then $m/n$ is rational.
III. If $m$ $∈$ $\mathbb Z$ and $n$ is rational, $m+n$ is rational.
I think they all are true, but that answer isn't in my guide, so i searched the flaw and i can't find it. Any help?
I. The only counter example is if $m =0$ then $mn = 0 \in\mathbb Q$.
But if $m \ne 0$ then $m = \frac pq; p, q\in \mathbb Z$ then if $mn = \frac rs; r, s \in\mathbb Z$ then $mn = \frac pqn = \frac rs$ so $n= \frac rs*\frac pq \in \mathbb Q$.
II. If $n = 0$ then $\frac mn$ is not defined. But if $n \ne 0$ and $m=\frac pq$ and $n = \frac rs$ then $\frac mn = \frac {\frac pq}{\frac rs} = \frac {ps}{rq}$. So ?
III. If $n = \frac rs$ then $m + \frac rs$ is .... what?