# 0=1 proof

1. Oct 16, 2005

### homology

I was surprised when a few months ago, while talking to a fellow student, he suggested that the way you prove an equality (like P=Q) is you start with P=Q and play with it until you get something that's true, then you "know" that P=Q is true.

Now this is rubbish of course, since a false premise can imply a true one. And I showed him the example:

2=1 subtract 1 from both sides
1=0 add 1 to the left and 2 to the right to get
2=2

But he scoffed and said, "sure, sure" but you're using what you're trying to prove (the fact that 2=1). Well its clear that I haven't made him a believer, I was wondering if any folks here had really juicy examples of trying to prove P=Q, a false statment and ending up with R=S, a true one.

Thanks a lot,

Kevin

2. Oct 16, 2005

### amcavoy

Let a=b

1 = 2

b = 2b

b = a+b

b(a-b) = a2-b2

ab-b2 = a2-b2

ab = a2

This is working backwards (since you wanted to start with a false statement) an example from MathWorld.

http://mathworld.wolfram.com/Fallacy.html

3. Oct 16, 2005

### HallsofIvy

Actually, that's a quite common method of proof, sometimes called "synthetic proof" and is often used in proving trigonometric identities: you start with identity (what you want to prove) and reduce it to something you know to be true.

homology is, of course, completely correct that a false statement can lead to a true one. You have to be careful that every step in a "synthetic proof" argument is reversible. What you are really doing is using a common method of deciding how to prove something- working backwards.
"Here is what I want to prove- what do I need to have so that that is clear? Okay, now what do I have to have in order to prove that?", continuing until you arrive at something you know to be true- a definition or axiom or a "given" part of the hypothesis. Having determined how to prove, you turn around and do everything in reverse- start with the "given" and work back to what you wanted to prove. As long as every step is reversible you can do that. If, in a simple proof, it is clear that every step is reversible, it may not be necessary to actually write out the "reverse" process- that's a "synthetic proof".

4. Oct 16, 2005

### homology

Hmm,with all due respect, it seems fishy to me. I have to say that I would never prove anything by first assuming it and then working from there. While I might play with such things on scrap paper, a final proof should start with what is known to be true and then by deduction arrive at the goal.

Could you direct me to a rigorous definition of synthetic proof?

Thanks,

Kevin

5. Oct 16, 2005

### Robokapp

Proving something is equal is much harder than proving it wrong. So...proving that 1=0

1^0=0^1 Wrong

1/0=0/1 Wrong

1-0=0-1 Wrong

(x-1)(x-0)=0
x^2-x-0=0 i'm completing the square
x^2-x-1/4-0+1/4=0
(x^2-x-1/4)+1/4=0
(x-1/2)^2+1/4=0
(x-1/2)^2=-1/4
(X-1/2)=+ and - 0.5i

so the equation formed by turning the given x values (0 and 1) into factors (x-1) and (x-0) does not have identical roots, therefore the factors must differ.

and you can keep on going like this as far as you want.

6. Oct 16, 2005

### eNathan

If you have

$$a = b$$

you cant add different numbers to the left and right, you have to add the SAME number, or subtract, multiply, divide...ect

you cant add 1 to the left and 2 to the right!

nice try, however.

7. Oct 16, 2005