Proofs

[rowshan]

what is the approach to questions that ask you to veryfy something or prove it.?I know there is that rhs = lhs method but i don't quite understand it. It all gets confusing when i have to use words. Gah.

[Collin Li]

Write out what you need to prove in a mathematical statement (translate the worded question), then prove it.

[shinny]

When it asks you to prove something, start from a deduction which can be made from the source material (which is usually something obvious), and sub in other stuff from the source material/plug in formulas until you get the required LHS. From what I know, this is the ONLY legit method of doing proofs. i.e. No working simultaneously on both sides, or using the result to prove the result.

If it says verify, then you can just sub in the result they provide and prove the statement which is provided without having to actually prove its derivation. If it says 'show that', I tend to opt for the proof method, but I'm not actually too sure whether you're allowed to use the verify method.

[Lycan]

Basically, with proofs in the form of an equation, you start with one side (Left or right), usually, the one that is more complicated is what I go with, because simplification is better than complicatingationismness, but either is fine depending on the situation. You are trying to get to what is on the other side. Of course, there are alternate methods like, starting with LHS, getting to a point, and then taking the RHS and getting to the same point.

For example, Prove that Blah = (Blah + 1 - 1)x1 if you start with the LHS.

LHS = Blah (given)
= Blah + 1 - 1
= (Blah + 1 - 1)x1

RHS = (Blah + 1 - 1)x1 (given) = LHS Therefore, proven.

What you don't want to do, is say something that amounts to 'because LHS = RHS, therefore, LHS = RHS'. As silly as it sounds, it happens often with the weaker students.

If the question asks you to verify Expression = Different looking expression, that is the same as a proof. If it's something like, verify that this is a solution of that. Then you probably know what to do.

Sorry if I misinterpreted the question.

When it asks you to prove something, start from a deduction which can be made from the source material (which is usually something obvious), and sub in other stuff from the source material/plug in formulas until you get the required LHS. From what I know, this is the ONLY legit method of doing proofs. i.e. No working simultaneously on both sides, or using the result to prove the result.

If it says verify, then you can just sub in the result they provide and prove the statement which is provided without having to actually prove its derivation. If it says 'show that', I tend to opt for the proof method, but I'm not actually too sure whether you're allowed to use the verify method.

I would advise the proof method when the question says 'show that'.

[Mao]

make sure you ALWAYS state what has been asked of you. doing a full proof without stating what it has proved won't attract full marks.

and remember to put "QED" for "prove that...." and "as required" for "show that..."/"verify that..." questions for added assurance.

[rowshan]

what does "QED" mean?

[Ahmad]

Quot erat demonstrandum - which was to be shown (proved). My personal preference is to use "as required".

Always keep in mind what a proof ultimately is. It is essentially an argument which will logically convince another person that some statement is true. No matter what method you use, if it does this correctly then your proof is correct.

Your teacher may give you a way to structure your proof which is fine. However, I think the main reason why they do this is because it's easy for people to incorrectly reason through a proof.

One of the pitfalls common to students is that they work from the end result towards a true statement blindly, for example,

Prove that .

If I worked from the end result like this:



(upon multiplying by cos squared).

which is a true statement. Can you spot why this isn't rigorous? (Note the implies sign, note this can be made rigorous).

The reason is you shouldn't work from the result to a true statement, unless every step is reversible! Because above I used an implies symbol, instead of an equivalence at each step. If that was all it took to prove something, then I could prove any equality you gave me (even a false one). Like this:

Prove .

Proof:

which is true, therefore my proof holds. (Wrong).

The problem here lies in the fact that we can't go from a true statement to the statement we're trying to prove, that is, does NOT imply .

So our initial proof should've actually been:

as required.

It may take you a while to see where I'm coming from because it's so natural to start from the end result towards a true statement. We do it all the time in algebra, when we solve things like . If you want to work backwards from the end result, you must show that the steps you take are reversible, something like this:

Proof:



(upon multiplying by cos squared).

which is true.

The equivalence arrow says that both statements are equivalent, therefore proving the 2nd statement is equivalent to proving the 1st statement. But we know the 2nd statement is true, therefore so is the 1st.

[rowshan]

 
I'm quite confused
Prove that sec^2(x) = tan^2(x) + 1.
Is this okay?
RHS=tan^2(x) + 1
=sin^2(x)/cos^2(X)+cos^2(x)/cos^2(x)
=(sin^2(x)+cos^2(x))/cos^2(x)
=1/cos^2(x)
=sec^2(x)
=LHS as required or QED
Is that it???

I have never used the implies nor have i seen it...or the equivalent, when am i supposed to use it?

"Proof:

\sec^2(x) = \tan^2(x) + 1

\Leftrightarrow \cos^2(x) + \sin^2(x) = 1 (upon multiplying by cos squared)."

The equivalence arrow says that both statements are equivalent, therefore proving the 2nd statement is equivalent to proving the 1st statement. But we know the 2nd statement is true, therefore so is the 1st."

But how do we know that cos^2(x) + sin^2(x) = 1? I know its true, but am i supposed to assume that the reader will know that this is true?

"One of the pitfalls common to students is that they work from the end result towards a true statement blindly, for example,"  Like i did? I went backwards?

I am so incredibly confused!

[Ahmad]

It's a quite subtle detail that you're probably not expected to know, and takes time to think about. Your proof would probably be ok, don't fret over it. I'm truly sorry if I confused you! If you're not sure about what I said, just stick to a specific method your teacher or book used, such as the LHS/RHS manipulation thing.

I'm pretty sure the pythagorean identity can be quoted, so you don't need to worry about that.

Note:

Implies is quite simple. If A and B are propositions, or things that can be either true or false, such as "it rained today", then means that if A is true, then B is true. The equivalence (double arrow) means that if one proposition is true, they're both true, and if one proposition is false, they're both false.

[rowshan]

Ahhh yes, but if you use the Implies sign that means that just because B is true doesn't necessarily mean that A is also true? In higher mathematics are you taught these logic things?

Thanks for the responses.

[Ahmad]

You're right there! And you should encounter some logic soon enough.

[bigtick]

One statement is true for all x, the other is true for some x. Are they equivalent?

[Ahmad]

I didn't want to be too confusing, but you're right. You should consider the domain and exclude where dividing by zero occurs.

Edit: In other words, they're equivalent for appropriately selected x.

[excal]

I like QED to mean 'Quite Enough Done'.

[phagist_]

Quantum Electrodynamics?