Random math questions

[#1procrastinator]

Yeah that's right, he proved exactly that by induction, usually the cleanest way of expressing it but not necessarily the most intuitive.

Ok, in the proof, the claim was for but then we wanted to show that it was true for , but how can we assume is true when that's what we're trying to prove in the first place?

an even number can be written in the form , if we can write each of first three terms in this form, then they are even:

Would you have to show that the sum of the even numbers is indeed even?

[polar]

true lol, I guess a better way would've been and hence, 1 plus an even number is odd

[QuantumJG]

These are some cool questions.

Hi #1procrastinator, maybe I should have used some better notation. I claimed that for any , I can write



So after doing the base case, I assume that my claim holds for some such that . Then I want to show that it also holds for (In your mind think of and ). Now I know that



Now I use my assumption and get



So now I've shown it holds for , and thus I've shown it holds in general because I could have made as arbitrarily large as I like.

Overall, this is just telescoping a series. In general (sometimes - if it's an infinite series, you should first see if it converges by doing convergence tests) for some sequence

[kamil9876]

Ok, in the proof, the claim was for but then we wanted to show that it was true for , but how can we assume is true when that's what we're trying to prove in the first place?

Read my most recent post for an explanation of what induction is, or just google "mathematical induction" to see what it is.

[#1procrastinator]

Thanks a lot guys - think I got it now - just gotta read up a bit more on this proof by induction witchery though.
Also, QuantumJG, in your last step:




Did you really have to take the limit into the exponent or could you just have said that is 1?

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1) What does 'coordinatewise addition' mean (e.g. 'Define addition of elements in V coordinatewise)? Does it just mean adding the corresponding components of two lists or vectors? So (a, b) + (c, d) = (a+c, b+d)

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2) To prove would this acceptable?



This is the way the author does it:




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3) To prove , he starts with the RHS and shows that it's equal to the LHS. Is this an example of a case where P<->Q? Generally, how would you determine if P<->Q or just P->Q?


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4) How would you write the set of all even functions using set notation?

[kamil9876]

Did you really have to take the limit into the exponent or could you just...

It really depends how much detail you want. Are you satisfied? Or maybe more importantly sometimes the question is are your bosses satisfied. If it was a 1st year course in calculus and you just covered continuity of exponential yesterday, then yeah I would. If however you were writing a research paper then you shouldn't, it would be a waste of paper.

1) What does 'coordinatewise addition' mean

Yeah it means do it on each coordinate. Again this is slang and if you are ever unsure about slang, look ahead and try to figure out the right definition from the context/examples etc.

2) To prove would this acceptable?

This depends entirely on your definition of , in most formal construction of say the rational numbers, we define the rationals as pairs of numbers where are integers with and define an equivalence relation by if and only if . If one was to use this definition then to show one would have to show that which obviously follows from commutatitivty and associativity of multiplication.

3) To prove \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bc}, he starts with the RHS and shows that it's equal to the LHS. Is this an example of a case where P<->Q? Generally, how would you determine if P<->Q or just P->Q?

I don't understand your concern. Which statements are P and Q supposed to represent in this particular example? As an exercise you may want to prove this identity using the definition of fractions I gave above.

4) How would you write the set of all even functions using set notation?

How would I write the set of all things that satisfy property P? . What is the here? Being a function f such that f(x)=f(-x) for all . So using this template



sometimes you can use : instead of | if it looks better (for example, when there are absolute values involved in my definitions I obviously prefer to use : )



A more economical and professional way of writing it would be:

[#1procrastinator]

^ Thank you!

This depends entirely on your definition of [/tex]

From the book ('Calculus' by Spivak), he says 'the symbol a/b means ab^-1.

I don't understand your concern. Which statements are P and Q supposed to represent in this particular example? As an exercise you may want to prove this identity using the definition of fractions I gave above.

I was thinking P would be on the LHS and Q would be the RHS but there's a damn equality sign there!

[kamil9876]

From the book ('Calculus' by Spivak), he says 'the symbol a/b means ab^-1.

Hrmm ok, then what are and assumed to be? Just elements of some field (like real numbers etc.) ?

So in that case proving means proving that . This follows from .

[#1procrastinator]

He doesn't say explicitly but pretty sure they're supposed to be real numbers :p

why do we have to write as ? And could we prove instead ?

Could you please point what is wrong with this way:


EDIT:

And also to prove that if , then if a is real but non-zero, is it sufficient to just do the following:

[kamil9876]

Look, when it comes to these sort of foundational things what counts as a proof and what doesn't largely depends on your definitions and what you already know¹  (i.e what facts/theorems are you allowed to use). So as for your second query, it depends... sure it is a good enough proof if you already know that the real numbers have multiplicative inverses.

why do we have to write a/b as ab^{-1}?

you told me that was the definition! so I used that definition, if you had given me another definition I would have used that definition. For such simple facts, might as well stick to the definitions and everything will work out.

[1] There are many paths one may take to develop the basics of the reals, I don't know which order Spivak goes in.

[TrueTears]

you should link arun's site about proofs here lol

[#1procrastinator]

Look, when it comes to these sort of foundational things what counts as a proof and what doesn't largely depends on your definitions and what you already know¹  (i.e what facts/theorems are you allowed to use). So as for your second query, it depends... sure it is a good enough proof if you already know that the real numbers have multiplicative inverses.

Well, I guess if I'm showing the proof to someone, I'm assuming they're familiar with the axioms I'm working with too :p

you told me that was the definition! so I used that definition, if you had given me another definition I would have used that definition. For such simple facts, might as well stick to the definitions and everything will work out.

[1] There are many paths one may take to develop the basics of the reals, I don't know which order Spivak goes in.

Haha, he mentions it almost in passing, saying that a/b always means ab^(-1) so I thought we could use either one.

you should link arun's site about proofs here lol

This?
http://www.ms.unimelb.edu.au/~ram/Notes/grammarContent.xhtml

[TrueTears]

ya that and another pdf document somewhere...

[#1procrastinator]

Show that if for all x, then


Not sure how to show this, some pointers would be nice :p

[kamil9876]

This is a linear algebra problem. Plug in three values of to get three linear equations. If lucky, these three equations should only have the trivial solution.

It may be efficient to plug in values that give zeros. For example:

gives .

gives

Now just need one more equation, I'll leave you to find this.

Alternatively, a more systematic way would be to plug in their Taylor series (if you already know what they are) to get some equations by comparing coefficients.