Proving Propositions: A Challenge!

[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.

 

[Dick]

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?

 

[guroten]

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.

 

[Dick]

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)?