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

• flyingpig
In summary, the conversation discusses a mathematical proof that can be done both forwards and backwards. The proof involves showing that for x>0, x+1/x ≥ 2 and the equality holds when x is 1. The proof is done using if and only if statements, but the TA suggests starting with a statement that is known to be true and working towards the desired conclusion. However, the speaker and another expert disagree with this approach and recommend starting with the desired conclusion and working backwards, as long as each step is reversible.

#### flyingpig

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

## Homework Statement

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??

Sorry I meant the title should be

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

flyingpig said:

## Homework Statement

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??

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.

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.

moxy said:
The way you did it makes it look like you started by assuming what you were trying to prove.

What does that mean?

flyingpig said:
If this was on an exam, how could I make up so much space and then erase and go back??

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.

flyingpig said:
What does that mean?

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.
micromass said:
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.

A pain, but a necessity!

I disagree with your TA.
You did it exactly right and the way I would recommend to do it.

I like Serena said:
I disagree with your TA.
You did it exactly right and the way I would recommend to do it.

Yes, the TA is wrong about this.

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!

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
$$x+ \frac{1}{x}\ge 2$$
$$x^2+ 1\ge 2x$$
$$x^2- 2x+ 1\ge 0$$
$$(x- 1)^2\ge 0$$.

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.

## 1. How can the proof be wrong if I do it backwards?

The proof can be wrong if you do it backwards because the steps in a proof are designed to build upon each other in a specific order. Doing the steps backwards can result in incorrect conclusions.

## 2. Why is the order of steps important in a proof?

The order of steps is important in a proof because each step relies on the previous step to be correct. Changing the order can lead to incorrect conclusions and invalidate the proof.

## 3. Can't I just rearrange the steps in a proof to make it easier to understand?

No, rearranging the steps in a proof can change the logic and invalidate the proof. The steps are carefully constructed to build upon each other and must be followed in the correct order to ensure the validity of the proof.

## 4. How do I know if I've done a proof backwards?

If you find yourself reaching an incorrect conclusion or encountering logical inconsistencies while doing a proof, it is likely that you have done it backwards. Double-checking the order of steps can help identify and correct any mistakes.

## 5. Why do we even need to do proofs in a specific order?

Proofs are a way to logically and systematically demonstrate the truth of a statement. Following a specific order ensures that the proof is valid and the conclusion is reliable. Without a specific order, the proof would lose its reliability and effectiveness.