Falsity of assumptions question.

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In summary, the conversation discusses the proof of the irrationality of √3 and the question of whether or not the proof is valid. The individual first assumes that √3 is rational and in its lowest form, and then goes on to prove that this leads to a contradiction. The question then arises if both assumptions are actually false or if only one could be false. It is clarified that the assumption of a rational number being in its lowest form is not necessary for the proof, but rather a statement made without loss of generality.
  • #1
tylerc1991
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In the course of proving that [itex] \sqrt{3} [/itex] is irrational, I had another question pop up. To prove that [itex] \sqrt{3} [/itex] is irrational, I first assumed 2 things: [itex] \sqrt{3}[/itex] is rational, and the rational form of [itex] \sqrt{3} [/itex] 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 [itex] \sqrt{3} [/itex] 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 [itex] \wedge [/itex] 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 [itex] \wedge [/itex] B being false.

On the other hand, the second assumption that was made (the rational form of [itex] \sqrt{3} [/itex] 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!
 
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  • #2
tylerc1991 said:
On the other hand, the second assumption that was made (the rational form of [itex] \sqrt{3} [/itex] 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?
 
  • #3
Hurkyl said:
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?
 
  • #4
Right, although I wouldn't have chosen that phrasing.
 
  • #5


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.
 

1. What is the definition of "falsity of assumptions question"?

A falsity of assumptions question refers to a type of question that challenges the validity or accuracy of underlying assumptions. It aims to reveal any faulty or unfounded assumptions that may have been made in a particular argument or statement.

2. How is a falsity of assumptions question different from a regular question?

A falsity of assumptions question is different from a regular question because it does not seek new information or clarification, but rather challenges the assumptions that have been made. Regular questions typically aim to gather information or seek understanding.

3. Why is it important to ask falsity of assumptions questions?

Asking falsity of assumptions questions is important because it allows for critical thinking and analysis of arguments and statements. It helps identify any flawed or biased thinking and promotes a more objective and logical approach to problem-solving.

4. When should falsity of assumptions questions be used?

Falsity of assumptions questions can be used in any situation where critical thinking and analysis are needed, such as in scientific research, debates, or decision-making processes. They are especially useful when evaluating the validity of arguments or when trying to uncover hidden biases or assumptions.

5. Can falsity of assumptions questions be used in any field of study?

Yes, falsity of assumptions questions can be used in any field of study. They are commonly used in science and philosophy but can also be applied in fields such as business, law, and psychology. Any situation that requires critical thinking and analysis can benefit from the use of falsity of assumptions questions.

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