# Proving Propositions: A Challenge!

• guroten
In summary, the conversation discusses proving propositions related to rational and irrational numbers, particularly the statement "If x is rational, then x + sqrt(2) is irrational." The individual has attempted various methods but has not been successful in proving the statement. They eventually find counterexamples for the other statements but struggle with finding one for the last proposition.
guroten

## Homework Statement

Prove these propositions

## Homework Equations

If x is rational, then x + sqrt(2) is irrational.
If x + sqrt(2) is irrational, then x is rational.
If xsqrt(2) is irrational, then x is rational.

## The Attempt at a Solution

I've tried a couple of ways, but I always end up at a dead end. usually, I end up assuming what I need to prove.

guroten said:

## Homework Statement

Prove these propositions

## Homework Equations

If x is rational, then x + sqrt(2) is irrational.
If x + sqrt(2) is irrational, then x is rational.
If xsqrt(2) is irrational, then x is rational.

## The Attempt at a Solution

I've tried a couple of ways, but I always end up at a dead end. usually, I end up assuming what I need to prove.

The first proposition is true. The others aren't. Can you find a counterexample for them? As to proving the first can you show us what you tried?

After working with it some more, I've gotten all but the last. Is a counter example something like e(sqrt(2))? But I don't know if that is rational or not.

guroten said:
After working with it some more, I've gotten all but the last. Is a counter example something like e(sqrt(2))? But I don't know if that is rational or not.

That's too hard. In fact, I'm not sure anybody knows the answer to that. How about sqrt(3)*sqrt(2)?

## 1. How do you prove a proposition?

Proving a proposition involves providing evidence or logical arguments that support the truth or validity of the proposition. This can be done through various methods such as deductive reasoning, inductive reasoning, or mathematical proofs.

## 2. What is the difference between a hypothesis and a proposition?

A hypothesis is a tentative statement or idea that is proposed to explain a phenomenon, whereas a proposition is a statement that is asserted as true or accepted as a principle.

## 3. Can a proposition be proven beyond a doubt?

No, it is not possible to prove a proposition beyond a doubt as there is always a possibility that new evidence or arguments may arise that challenge its validity. However, a proposition can be proven to a high degree of certainty based on the available evidence and logical reasoning.

## 4. How do you know if a proposition is true?

The truth of a proposition can be evaluated based on the evidence and logical arguments presented in its support. It is also important to consider any potential biases or limitations in the evidence and to remain open to new information that may challenge the proposition.

## 5. What is the role of experimentation in proving propositions?

Experimentation can be a valuable tool in proving propositions, especially in the scientific method. By conducting controlled experiments, researchers can gather empirical evidence to support or refute a proposition. However, not all propositions can be proven through experimentation, and other methods of proof may be necessary.

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