# Help choosing best way to prove logic problem

• Feb 28th 2010, 09:20 AM
piyourface166
Help choosing best way to prove logic problem
Ok I don't need answers just a little shove in the right direction. It's not an incredibly hard logic problem, I just want to know the best route to take to prove it. Here's the question,

If 'R' and 'S' are irrational then R^S is irrational.

I was thinking contra-positive would be best. So assume not P and prove not Q. So this means I would have 'R' and 'S' are rational and prove that R^S is rational. I know that if a number is rational then it can be represented by two integers (m/n) but I'm just not sure how to work that in to get my proof solid. Thanks in advanced.
• Feb 28th 2010, 09:29 AM
Drexel28
Quote:

Originally Posted by piyourface166
Ok I don't need answers just a little shove in the right direction. It's not an incredibly hard logic problem, I just want to know the best route to take to prove it. Here's the question,

If 'R' and 'S' are irrational then R^S is irrational.

I was thinking contra-positive would be best. So assume not P and prove not Q. So this means I would have 'R' and 'S' are rational and prove that R^S is rational. I know that if a number is rational then it can be represented by two integers (m/n) but I'm just not sure how to work that in to get my proof solid. Thanks in advanced.

I completely disagree. Let $R=e,S=\ln(2)$. Both are irrational but $R^S=2$
• Feb 28th 2010, 10:00 AM
piyourface166
That's a good example. So I know that we have AT LEAST one example, I'm sure more, than we can prove it is true. My problem still is what is my best bet to write a proof for this. If the question was to show and example or prove this true or false for one example that would work. My issue is I'm not sure what to do for my next step in proving that for 'the universe' of all irrational numbers this is true.

My point is I can show that there is a false one to, for example:
(Root2 ^ Root3) is going to be irrational.

You showed an example that makes two irrationals a rational number, so obviously this problem is going to be false because that right there is a contradiction. What do you think the best method of proving would be to finish this? P and not Q, then show an example???
• Feb 28th 2010, 10:15 AM
piyourface166
I got it, thanks!
• Feb 28th 2010, 01:29 PM
Defunkt
Quote:

Originally Posted by piyourface166
That's a good example. So I know that we have AT LEAST one example, I'm sure more, than we can prove it is true. My problem still is what is my best bet to write a proof for this. If the question was to show and example or prove this true or false for one example that would work. My issue is I'm not sure what to do for my next step in proving that for 'the universe' of all irrational numbers this is true.

My point is I can show that there is a false one to, for example:
(Root2 ^ Root3) is going to be irrational.

You showed an example that makes two irrationals a rational number, so obviously this problem is going to be false because that right there is a contradiction. What do you think the best method of proving would be to finish this? P and not Q, then show an example???

I don't understand. Drexel28's example obviously shows this claim to be false, hence there can be no valid proof for it! The fact that you were asked to prove a wrong claim does not make it right (or provable).

By the way, you have your contra-positive the wrong way. If you want to prove $p \Rightarrow q$ by contra-positive, you need to prove that if q doesn't hold then neither does p; that is, in this case, you would assume that $R^S$ is rational, and show that either $R$ or $S$ are rational as well.