# Do they allow you to use “reversing the step” in the HSC? (Nature of proof) (1 Viewer)

#### catha230

##### New Member
Allow me to elaborate on the title. So the new syllabus has introduced a new topic called “nature of proof” which basically consists of proving problems. At tutor, I was told that according to NESA, you are not allowed to assume something is true and work backwards to the end. However, at school when doing a proving question, my teacher often just write what we're required to prove first and do it backwards until we reach something that is obviously true such as 1>0. Then at the end of the step, he writes "Reverse the steps to get the proof required". I'm really perplexed by that and wonder who I should listen to. I will attach a handwritten example on my school teacher's method to further clarify my query.

Thank you so much in advance!

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#### Drdusk

##### π
Moderator
Allow me to elaborate on the title. So the new syllabus has introduced a new topic called “nature of proof” which basically consists of proving problems. At tutor, I was told that according to NESA, you are not allowed to assume something is true and work backwards to the end. However, at school when doing a proving question, my teacher often just write what we're required to prove first and do it backwards until we reach something that is obviously true such as 1>0. Then at the end of the step, he writes "Reverse the steps to get the proof required". I'm really perplexed by that and wonder who I should listen to. I will attach a handwritten example on my school teacher's method to further clarify my query.

Thank you so much in advance!
Yeah you are not allowed to do that. Quote from my tutor when I was in 4u "Assumptions that are mathematically incorrect CAN lead to results that are mathematically correct."

Instead what you should do if you really want to work backwards is use a Proof by Contradiction. So instead start by assuming

$\bg_white \sqrt{1 + x} \geq 1 + \dfrac{x}{2}$

$\bg_white \dfrac{x^2}{4}\leq 0$

Which is obviously false giving a contradiction. Hence your assumption is false.

However what your teacher is saying I think is do it by assuming its true so you figure out what you need to start with in order to reach the answer. From this then rub out all the steps and write them in reverse. This IS allowed. However you can not just leave the proof like that, you must write the steps in reverse order. It's sort of a 'cheat' method, it works but you MUST remember to cross it out or else you'll lose marks.

#### fan96

##### 617 pages
At tutor, I was told that according to NESA, you are not allowed to assume something is true and work backwards to the end. However, at school when doing a proving question, my teacher often just write what we're required to prove first and do it backwards until we reach something that is obviously true such as 1>0.
Whether or not you can work backwards depends on if the steps you take are invertible/reversible, or in other words, the implications you use are of the "if and only if" type.

For example,

\bg_white \begin{aligned} x+ 5 &= 0\\ \iff x &= -5 \end{aligned}

is an example of such an invertible operation, because $\bg_white x+ 5 = 0$ is logically equivalent to $\bg_white x = -5$ - each statement implies the other.

Here you can start from the statement $\bg_white x = -5$ and work backwards to get $\bg_white x + 5 = 0$. Then you may conclude that if $\bg_white x + 5 = 0$, then $\bg_white x = -5$.

However, the $\bg_white \sin$ function is not invertible.

\bg_white \begin{aligned} x &= \pi \\ \implies \sin x &= 0 \end{aligned}

The first statement implies the second, but the second does not imply the first.

So in this case you could not start from $\bg_white \sin x = 0$ and then conclude that $\bg_white x = \pi$.

You should certainly never assume something is true. So your tutor is correct, but what your teacher is doing is not assuming the statement is true. Rather, they are just finding logically equivalent statements, continuing this process until one of these equivalent statements is true, which proves the original statement.

Now in general the first two are not invertible, but since all the quantities involved are positive, this is okay.
Of course, addition is always an invertible operation, so this proof is valid.

The key here is ensuring that the steps you take can be done in either direction.

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#### catha230

##### New Member
Yeah you are not allowed to do that. Quote from my tutor when I was in 4u "Assumptions that are mathematically incorrect CAN lead to results that are mathematically correct."

Instead what you should do if you really want to work backwards is use a Proof by Contradiction. So instead start by assuming

$\bg_white \sqrt{1 + x} \geq 1 + \dfrac{x}{2}$

$\bg_white \dfrac{x^2}{4}\leq 0$

Which is obviously false giving a contradiction. Hence your assumption is false.

However what your teacher is saying I think is do it by assuming its true so you figure out what you need to start with in order to reach the answer. From this then rub out all the steps and write them in reverse. This IS allowed. However you can not just leave the proof like that, you must write the steps in reverse order. It's sort of a 'cheat' method, it works but you MUST remember to cross it out or else you'll lose marks.
so in the hsc, simply writing “reverse the steps to get the proof” is not enough? Does it mean that I need to re-write the whole thing? If so, it’s gonna be really time consuming. I reckon I’ll use contradiction as you told. Thank you so much for your reply

#### catha230

##### New Member
Whether or not you can work backwards depends on if the steps you take are invertible/reversible, or in other words, the implications you use are of the "if and only if" type.

For example,

\bg_white \begin{aligned} x+ 5 &= 0\\ \iff x &= -5 \end{aligned}

is an example of such an invertible operation, because $\bg_white x+ 5 = 0$ is logically equivalent to $\bg_white x = -5$ - each statement implies the other.

Here you can start from the statement $\bg_white x = -5$ and work backwards to get $\bg_white x + 5 = 0$. Then you may conclude that if $\bg_white x + 5 = 0$, then $\bg_white x = -5$.

However, the $\bg_white \sin$ function is not invertible.

\bg_white \begin{aligned} x &= \pi \\ \implies \sin x &= 0 \end{aligned}

The first statement implies the second, but the second does not imply the first.

So in this case you could not start from $\bg_white \sin x = 0$ and then conclude that $\bg_white x = \pi$.

You should certainly never assume something is true. So your tutor is correct, but what your teacher is doing is not assuming the statement is true. Rather, they are just finding logically equivalent statements, continuing this process until one of these equivalent statements is true, which proves the original statement.

Now in general the first two are not invertible, but since all the quantities involved are positive, this is okay.
Of course, addition is always an invertible operation, so this proof is valid.

The key here is ensuring that the steps you take can be done in either direction.
Hi, thanks for your interest in my thread. Does that mean that in the HSC it is simply not enough just to write “reverse the steps to get the proof”?

#### Drdusk

##### π
Moderator
so in the hsc, simply writing “reverse the steps to get the proof” is not enough? Does it mean that I need to re-write the whole thing? If so, it’s gonna be really time consuming. I reckon I’ll use contradiction as you told. Thank you so much for your reply
Yeah you must re-write the whole thing or you can lose marks. It is time consuming but sometimes it actually helps believe it or not. Some questions are really hard and you just don't know where to start so by working backwards you can figure it out. However yes first try the Proof by Contradiction and other normal ways of proving it and if you can't get it out then work backwards.

#### fan96

##### 617 pages
Hi, thanks for your interest in my thread. Does that mean that in the HSC it is simply not enough just to write “reverse the steps to get the proof”?
For the proof to be mathematically valid, you need to show that these steps actually are reversible. Then the proof is correct.

But I don't know how an HSC marker would react to this - this sort of logic wasn't touched on at all in the old HSC. But it is covered in first year uni math so I would expect most markers to accept an explanation of why the steps are reversible.

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#### stupid_girl

##### Active Member
I think "Reverse the steps to get the proof required" actually means you have to write the whole stuff again in reversed order.

The technique of working backwards is extremely useful when you learn how to prove limit in university. (It would be very difficult to think of a correct bound otherwise.) However, ultimately you still need to write the proof in correct order to get marks.

#### Drdusk

##### π
Moderator
The technique of working backwards is extremely useful when you learn how to prove limit in university. (It would be very difficult to think of a correct bound otherwise.) However, ultimately you still need to write the proof in correct order to get marks.
Don't even get me started on that. My friends and I have been arguing about the Epsilon Delta crap for 3 trimesters now.

#### TheOnePheeph

##### Active Member
I think "Reverse the steps to get the proof required" actually means you have to write the whole stuff again in reversed order.

The technique of working backwards is extremely useful when you learn how to prove limit in university. (It would be very difficult to think of a correct bound otherwise.) However, ultimately you still need to write the proof in correct order to get marks.
When you say this do you mean like proving that the limit of a function at a certain x, a, is a certain value, l (l and a are both given), by showing that for all positive epsilon, a positive value of delta exists so that for all x, if |x-a|<delta then |f(x)-l|<epsilon?

#### stupid_girl

##### Active Member
When you say this do you mean like proving that the limit of a function at a certain x, a, is a certain value, l (l and a are both given), by showing that for all positive epsilon, a positive value of delta exists so that for all x, if |x-a|<delta then |f(x)-l|<epsilon?
Yes. It's very difficult (or nearly impossible) to choose a delta without working backwards on a rough work paper.

#### TheOnePheeph

##### Active Member
Yes. It's very difficult (or nearly impossible) to choose a delta without working backwards on a rough work paper.
Yeah I see what you mean. I don't even see how you could do it without working backwards from an assumed limit.

#### InteGrand

##### Well-Known Member
Hi, thanks for your interest in my thread. Does that mean that in the HSC it is simply not enough just to write “reverse the steps to get the proof”?
$\bg_white \noindent You could in theory add the \color{blue}\Leftrightarrow (or even \color{blue}\Leftarrow) symbol before each line after the first line to make the proof valid. However, I'm not sure if the HSC markers would accept it.$

#### no_arg

##### Member
Consider the following

Theorem: -2=2.

Proof: -2=2 implies that |-2|=|2| and hence 2=2 which is true.

Therefore -2=2.

Logic, like water, usually only flows in one direction.

Even in the old days, HSC markers were brutal on responses which ran in the wrong direction, particularly inequality proofs.
Maaaaaaaaaannnnny looooooooooooost marks.
You should not start the proof of a theorem with the theorem.

Cheers

#### no_arg

##### Member
Consider the following

Theorem: -2=2.

Proof: -2=2 implies that |-2|=|2| and hence 2=2 which is true.

Therefore -2=2.

Logic, like water, usually only flows in one direction.

Even in the old days, HSC markers were brutal on responses which ran in the wrong direction, particularly inequality proofs.
Maaaaaaaaaannnnny looooooooooooost marks.
You should not start the proof of a theorem with the theorem.

Cheers
With regard to epsilon-delta arguments remember that lines of a proof do not need justification.
They just needs to be true.
If your proof starts with the statement 0<1 there is no need to agonise over why you started there..........that is your choice.

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