# Problem with formulating the negation of a statement

The excersise asks us to prove with an Indirect Proof that if $$abc$$ is an irrational number then at most two of $$a,b,c$$ are rational numbers.

So we have,

Let $$P$$ denote "$$abc$$ is irrational" and let $$Q$$ denote "at most two of $$a,b,c$$ are rational numbers". We want to prove: If $$P$$ then $$Q$$, but indirectly, i.e. $$\neg P => \neg Q$$. Formulating the negation of $$P$$ we have

$$\neg P :=$$ "$$abc$$ is rational"

But i don't know how to formulate the negation of $$Q$$. I figured that it should be "at least three of $$a,b,c$$ are rational numbers", but then $$\neg P$$ doesn't imply $$\neg Q$$ because we could have $$a=b=\sqrt2$$ and $$c=1$$ and so

$$abc=1\sqrt2 \sqrt2 =2$$

which is a rational number even though $$a,b$$ are irrational and clearly, this disproves $$\neg P => \neg Q$$. How should I formulate $$\neg Q$$? And in general, if we have more than one conditions in a statement, which conditions become negatives?

First of all $$P$$ implies $$Q$$ is not the same as $$\neg P$$ implies $$\neg Q$$. It is $$\neg Q$$ implies $$\neg P$$.
The negation of $$Q$$ is 'all three of $$a,b,c$$ are rational'.
The negation of $$P\implies Q$$ is $$P$$ and not $$Q$$ Thus you need to assume that $$abc$$ is irrational and all three of them are rational and derive a contradiction out of that.