# How come the proof is wrong if I do it backwards?

1. Nov 14, 2011

### flyingpig

How come the proof is "wrong" if I do it backwards?

1. The problem statement, all variables and given/known data

Let x >0. Then show that

$$x + 1/x \geq 2$$

and that the equality holds when x is 1

I got full marks on this, but remarked by my TA that I should do it backwards next time

Proof

$$x + 1/x \geq 2 \iff x^2 + 1 \geq 2x \iff x^2 - 2x + 1 \geq 0 \iff (x - 1)^2 \geq 0$$

Also the equality is true if x = 1

$$(1 - 1)^2 = 0^2 \geq 0$$

Q.E.D

My TA said I should start with $$(x - 1)^2 \geq 0$$ and go backwards. Why? If this was on an exam, how could I make up so much space and then erase and go back??

2. Nov 14, 2011

### flyingpig

Re: How come the proof is "wrong" if I do it backwards?

Sorry I meant the title should be

How come the proof is "correct" if I do it backwards?

3. Nov 14, 2011

Re: How come the proof is "wrong" if I do it backwards?

You won't need to write it down and then erase it and go back. You'll simply use the symbol $\iff$ to indicate that you can do it backwards. If all the steps you do in a proof are also valid in the reversed order then you can do the reasoning backwards. Can't you? the main point is that all those statements are "if and only if" statements, so they are logically equivalent.

4. Nov 14, 2011

### moxy

Re: How come the proof is "wrong" if I do it backwards?

Since you used iff arrows, I don't see anything wrong with your proof. However, I think your TA just wanted you to start by writing something that you know is true (i.e. (x+1)2 ≥ 0) and then work towards what you're trying to prove. The way you did it makes it look like you started by assuming what you were trying to prove.

But like I said, the statements are iff, so I don't really see a problem.

5. Nov 14, 2011

### flyingpig

Re: How come the proof is "wrong" if I do it backwards?

What does that mean?

6. Nov 14, 2011

### micromass

Staff Emeritus
Re: How come the proof is "wrong" if I do it backwards?

Write a draft first on a separate peace of paper. Once you did that, you know how much space you need and you can make a clean copy.

7. Nov 14, 2011

### moxy

Re: How come the proof is "wrong" if I do it backwards?

You should never start a proof by assuming that what you're trying to prove is true. So you shouldn't do this:

Claim: When x > 0, x+1/x ≥ 2

Proof: Let x>0. Assume x+1/x ≥ 2 ...

If your TA wasn't paying attention and missed that you used if and only if statements, he might have thought that you were assuming the claim was true and going from there.

A pain, but a necessity!

8. Nov 14, 2011

### I like Serena

Re: How come the proof is "wrong" if I do it backwards?

You did it exactly right and the way I would recommend to do it.

9. Nov 14, 2011

### micromass

Staff Emeritus
Re: How come the proof is "wrong" if I do it backwards?

However, I always tell my students not to use $\Leftrightarrow$. It's too dangerous. It's easy to say two things are equivalent when they are really not. I always ask my students to construct two proofs: one in the forward and one in the backward direction. This eliminates a lot of mistakes. It's a personal preference though and I would certainly not mark it incorrect!!

10. Nov 15, 2011

### HallsofIvy

Staff Emeritus
Re: How come the proof is "wrong" if I do it backwards?

You certainly can use "synthetic proof" (Starting with what you want to prove and ending with an obviously true statement) as long as you are careful that every step is "reversible". As long as it is given that x> 0, there is nothing wrong with
$$x+ \frac{1}{x}\ge 2$$
$$x^2+ 1\ge 2x$$
$$x^2- 2x+ 1\ge 0$$
$$(x- 1)^2\ge 0$$.

11. Nov 15, 2011

### I like Serena

Re: How come the proof is "wrong" if I do it backwards?

I like to start in a case like this with what you want to proof, and use single arrows to a true statement.

Afterward, convert each single arrow to a double arrow, while checking if it is legal to do so.

This way, it is clear to a reader (and yourself) why and how the proof starts, and he can verify for himself if the reverse implications hold.