ATAR Notes: Forum

Uni Stuff => Universities - Victoria => University of Melbourne => Topic started by: Deleted User on March 26, 2013, 11:05:11 am

Title: Accelerated Maths Help!!!
Post by: Deleted User on March 26, 2013, 11:05:11 am: I am so confused about the new topic mathematical induction. Can someone explain the steps required to prove something? And what basic assumptions do I need to know?
Title: Re: Accelerated Maths Help!!!
Post by: Leronziia on March 26, 2013, 01:08:07 pm: Not taking accelerated mathemativs but did this topic in the IB. First you have a proposition, anything, suppose the sum of 1 + 3 + 5 + 7 + ...... (2n - 1) = whatever.

Next we show that n = 1 is true, so when n = 1 the LHS is 1, suppose the RHS is also 1, then this only proves that n = 1 is true.

Next we must assume the proposition is true for any integer k, that is: 1 + 3 + 5 + 7 + ..... (2k - 1) = whatever. (change all n to k)

Now comes the fun part, you need to show that the (k + 1)th result is true (and therefore the proposition is always true) by adding the (k + 1)th term to both sides (2(k + 1) - 1). Then rearrange the RHS with your original 'whatever' term and the (2(k + 1) - 1) so that you express the RHS as the 'whatever' term except that every k now becomes (k + 1).

Your proposition is now proven, but this adding the (k + 1)th term to both sides works only with proving sums, every different proof requires its own special method which you need to sort of remember.

I tried my best explaining the process by text :P

Watch some videos of PatrickJMT doing it on youtube buddy :)
Title: Re: Accelerated Maths Help!!!
Post by: nubs on March 26, 2013, 02:45:37 pm: Maybe if you gave us a specific example, we could guide you through the process
Title: Re: Accelerated Maths Help!!!
Post by: Deleted User on March 26, 2013, 06:38:15 pm: Thanks Leronz!

This is the first question I have to do:

If a, b, a1, b1, c and d are integers, and if a/b=a1/b1, then a/b + c/d = a1/b1 + c/d and (a/b)*(c/d) = (a1/b1)*(c/d).

It seems very redundant and I always feel like I'm going in circles when trying to do it.
Title: Re: Accelerated Maths Help!!!
Post by: Leronziia on March 26, 2013, 06:54:23 pm: what exactly do we have to prove there? it seems like just a bunch of definitions/identities which obviously make sense mathematically.
Title: Re: Accelerated Maths Help!!!
Post by: kamil9876 on March 26, 2013, 07:35:49 pm: Quote
Thanks Leronz!

This is the first question I have to do:

If a, b, a1, b1, c and d are integers, and if a/b=a1/b1, then a/b + c/d = a1/b1 + c/d and (a/b)*(c/d) = (a1/b1)*(c/d).

It seems very redundant and I always feel like I'm going in circles when trying to do it.

What are your definitions of fractions, fraction addition and multiplication? The point of this particular exercise is most likely to show that the operations are well defined i.e don't depend on which fraction you choose.

$\mathbb{Q}$ is defined as the set of pairs $\frac{a}{b}$ where $a,b$ are integers with $b$ not zero. We identify certain pairs, declare $\frac{a}{b}=\frac{c}{d}$ if $ad-bc=0$ , so technically the rationals are a set of equivalence classes of pairs.

We then define $\frac{a}{b} + \frac{c}{d}=\frac{ad+bc}{bd}$ but the issue is how do you know that if $\frac{a}{b}=\frac{a'}{b'}$ and $\frac{c}{d}=\frac{c'}{d'}$ then is the sum well defined? i.e is the following true:

$\frac{ad+bc}{bd}=\frac{a'd'+b'c'}{b'd'}$

A priori we don't know that it is i.e when you randomly make up some mathematical object and define some operations, it may be the case that these operations aren't well defined i.e the operations depend on how you represent the elements. So let's check this, according to our definition of equivalence of fractions we must check that the following is $0$ :

$<br />(ad+bc)b'd' - (a'd'+b'c')bd = adb'd'+bcb'd' - bda'd' - bdb'c' = dd'(ab'-ba') + bb'(cd'-dc') = dd'.0+bb'.0=0$

Where we got the fact that $(ab'-ba')$ and $(cd'-dc')$ are 0 is because of the equivalence $\frac{a}{b}=\frac{a'}{b'}$ and $\frac{c}{d}=\frac{c'}{d'}$ respectively.

So the point is: Use the definitions of fractions given, only use properties that you know about integers (whole numbers). This is really an exercise in verifying that the construction of fractions is kosher and that they do indeed behave the way we expect from primary school.
Title: Re: Accelerated Maths Help!!!
Post by: Deleted User on March 26, 2013, 07:47:59 pm: Quote from: Leronziia on March 26, 2013, 06:54:23 pm
what exactly do we have to prove there? it seems like just a bunch of definitions/identities which obviously make sense mathematically.

Sorry! Prove that the equation I wrote is true.
Title: Re: Accelerated Maths Help!!!
Post by: Deleted User on March 26, 2013, 08:09:45 pm: Thanks for the response Kamil. I understand what you said, but I wouldn't know how to approach a different question. Are there steps that I can take when trying to do these sort of questions? It all seems very abstract to me.

Also, could you explain how to prove the second part? I keep going around in circles.
Title: Re: Accelerated Maths Help!!!
Post by: kamil9876 on March 26, 2013, 10:47:00 pm: Well you have to understand what it is that you are doing in your course. What definitions did the lecturer use? What do you already know? etc. Then just use those definitions. There are many ways of building up the foundations of maths and I don't know which path your course is taking so I can't say.
Title: Re: Accelerated Maths Help!!!
Post by: Deleted User on March 27, 2013, 10:53:20 am: I didn't really understand the definitions he gave it. But could you quickly explain to me how to prove the second part of the question (the multiplication part)? I think I'm starting to understand it