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.
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.
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.
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.
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.
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.
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.