Falsity of assumptions question.

[tylerc1991]

In the course of proving that $\sqrt{3}$ is irrational, I had another question pop up. To prove that $\sqrt{3}$ is irrational, I first assumed 2 things: $\sqrt{3}$ is rational, and the rational form of $\sqrt{3}$ is in it's lowest form. I then broke the proof up into cases and showed that none of these cases could occur.

My question boils down to: did I actually show that $\sqrt{3}$ is irrational?

From a purely logical standpoint, let's say that the 2 assumptions I made were named A and B. I successfully showed that A $\wedge$ B is false. However, this doesn't mean that BOTH A and B are false. More specifically, A could be true and B could be false, and I would still arrive at A $\wedge$ B being false.

On the other hand, the second assumption that was made (the rational form of $\sqrt{3}$ is in it's lowest form) shouldn't (doesn't?) change the problem.

Could someone give me solace and explain this little technicality I have? Thank you very much!

 

[Hurkyl]

On the other hand, the second assumption that was made (the rational form of $\sqrt{3}$ is in it's lowest form) shouldn't (doesn't?) change the problem.

If you are merely assuming that, then you do have a problem, and you have merely proved:

If sqrt(3) is rational, then it cannot be expressed as a fraction in lowest terms​

(or something equivalent)

But you don't have to merely assume that a rational number can be written in lowest terms, do you?

 

[tylerc1991]

But you don't have to merely assume that a rational number can be written in lowest terms, do you?

So this isn't really an assumption, per se? This is more of a 'without loss of generality' statement?

 

[Hurkyl]

Right, although I wouldn't have chosen that phrasing.

 

[blue_raver22]

I can understand your concern about the validity of your proof. However, I can assure you that your proof is still valid and that you have indeed shown that \sqrt{3} is irrational.

Firstly, let's address your concern about the assumptions A and B. In mathematics, when we make assumptions, we are essentially setting up a hypothetical scenario in order to prove or disprove a statement. In your case, you assumed that \sqrt{3} is rational and in its lowest form. By showing that both of these assumptions lead to a contradiction, you have effectively proven that the statement " \sqrt{3} is irrational" is true.

Secondly, the assumption that the rational form of \sqrt{3} is in its lowest form is a necessary one in order to prove the irrationality of \sqrt{3}. This is because if we allow for the possibility that \sqrt{3} can be expressed in a simpler form, then it is no longer irrational. So, this assumption does play a crucial role in the proof.

In conclusion, your proof is valid and you have successfully shown that \sqrt{3} is irrational. It is important to remember that in mathematics, we use assumptions to set up hypothetical scenarios and by showing that these assumptions lead to a contradiction, we prove the original statement to be true. So, you can take solace in the fact that your proof is sound and your conclusion is correct.