Help with Proof by Contradiction

Yeah I'm asking because I'm doing the STEP papers and the only formal proof I learn on Edexcel if proof by induction and they ask you about other methods of proving stuff on the STEP exams

1. Have a look here.

2. Have a read here.

3. [URL="http://docs.google.com/viewer?a=v&q=cache:hRzGM03P8fQJ:cheng.staff.shef.ac.uk/proofguide/proofguide.pdf+proofs+filetype[excludedFace]tongue[/excludedFace]df&hl=en&gl=uk&pid=bl&srcid=ADGEESgxgLVlM61hDjwss6m95TZexOA4dVzngw1rI8fly2wBa3-hjwznEbqlaWI7nPb3KguvVHHWLUxCtqlsjwxFpXNc1_H9g7A6on19xAUKSehlFhDPz4Izo5tqOPF1Fr7V8SsDmnS2&sig=AHIEtbRe6AbMD5-kR9wjaedVu9v0fcfvdA"]More to read here

$\Rightarrow 2b - a = 2k$ for some $k \in \mathbb{Z}$

$...$


$\therefore hcf(a, b) \not= 1$

Could you please clarify what you did there? When I substitute $2b-2k$ for $a$, I end up with $b=k$.

How does that lead to $\therefore hcf(a, b) \not= 1$?

This is NOT a proof of anything, so don't even try to follow it. I should have made it clearer.

Well, it's not a proof because there's no contradiction. However, Hasufel's proof has a clear contradiction: a and b are both even, therefore they have a common factor. What's wrong with that proof?

Perhaps you did notice that I reduced my statement to the statement of Hasufel?

What you are saying has nothing to do with my proposition.

Hence, by your argument $2 - \sqrt{4} = 2 - 2 = 0$ is an irrational number.

And what exactly is your proposition? I'm now quite confused as you can probably tell.



Should I ignore this too? Whose argument are you referring to?

I don't see why we should waste more time with this?

$2 - \sqrt{16} = \frac{a}{b}, hcf(a, b) = 1$

$\Rightarrow 2b - a = b\sqrt{16}$

$\Rightarrow (2b - a)^2 = 16b^2$

$\Rightarrow 2b - a = 2k, k \in \mathbb{Z}$

.. from now on you can continue with the proof given above.


What I'm asking is, what makes it different? Probably that you know in advance that $\sqrt{2}$ is irrational? (what you pretend you aren't assuming)

Reduce it to contradiction and let me know.