What is the contradiction of the statement $\forall r\in\mathbb{R}\ (\mathit{irrational}(r^3)\Rightarrow \mathit{irrational}(r))$?

Question

I have a question that asks me to prove by contradiction this expression:

$\forall r\in\mathbb{R}\ (\mathit{irrational}(r^3)\Rightarrow \mathit{irrational}(r))$

Now my understanding of a proof by contradiction is that you assume that the opposite of the claim is true. However, is this the same as negating it? If it were being negated, then my understanding is that it would be:

$r^3$ is irrational and $r$ is rational

based on the negation of implication. I'm not really sure what this means in terms of a proof by contradiction. Should it actually mean that we should assume that if $r^3$ is irrational, then $r$ is rational?

Answer 1

The negation of you statement is that $\exists\ r \in \Bbb R\ $ such that $rational(r)$ and $irrational(r^3)$ are both true simoultaneously.

You should suppose that such a number $r$ exists and that should lead you to a contraddiction