Fun Stuff > CHATTER
Can't Think of a Breaking Bad Pun For the Title: Let's Do Some Math!
snalin:
If you can't differentiate, can't you just draw the sets? Like A, that's just the curve of the function f(x)=x+(1/x) in the range (0, 5). Then you can just circle the stuff you are looking for. "Here, it's here. You can see it on the drawing". Since you're working within a limited domain, that's a valid proof.
I'll give some hints - though I do not guarantee their correctness, and I use the definitions of infimum and supremum from wikipedia, as I've never seen those terms before.
So for A, the minimum is 2, and the infimum is also 2 - if you draw it as a curve, the function goes from infinity at 0 to 2 at 1, to 5.2 at 5. Thus all values in A are >= 2. The maximum is infinity and the supremum is also infinity.
B is the union of [1,3] and [-1,-3] (you might need to prove that? I dunno). Minimum is -3, of course. For any number n<-3, n^2>9, as x^2 is a decreasing function for x<0, so the infimum is -3. The reverse goes for maximum and supremum of 3.
C is [1, 16) - show that for 0<x<1 and x >= 16, there is no y in [1,4) such that y^2=x. (and mention that the same goes for x<=0, but that's trivial)
D is {-1, 1/2, -1/3, 1/4, -1/5 ...}. Proofs will have to rely on the fact that the absolute value of elements are decreasing, so there's no x in D such that x < -1 or x > 1/2
E is all positive numbers, at least, otherwise that's too much for me to get into after a day of uni work.
ankhtahr:
Sadly having drawn it won't suffice. We have to give the professor a mathematical proof of our statements, either by mathematical induction, proof by exhaustion or reductio ad absurdum.
This is too damn difficult for the first exercise sheet.
By the way, everything I'm allowed to use is written in this German script up to page 19.5 (page 28.5 of the PDF):
Inofficial script
Mlle Germain:
Hi Ankhtahr,
I don't have much time, so I won't try to solve those exercises right now (I probably also would have difficulties doing it, since once you have proved all the basic stuff in maths, you kind of just use most of them without remembering how exactly the proofs work on an elementary level - just like you did before having seen the formal treatment). But I think we come from the same part of Germany, so our curriculum at school would have been similar. So I just wanted to encourage you by saying that in my experience, (almost) nobody who does maths at uni knows how to do proofs at the beginning and that people from other parts of Germany did not seem to have much of an advantage over me, although they might have seen sequences and series and such things before. I was very confused and annoyed about not knowing how proofs work during my first weeks of maths lectures, but after having seen more proofs in the lectures, you get used to the methods and to the way of thinking. Really don't worry, even if some people seem to understand everything and find it really easy, either they're just pretending or it will level out quite quickly. Uni maths just comes as a bit of a shock, since it's so different from school.
I hope you're going to love your maths lectures after a while; logical thinking and formal proofs are awesome! (That said, I'm going back to my differential geometry sheet...)
NotAwesomeAnymore:
Ankhtahr, it sounds like you're doing an introduction to Real Analysis. Real Analysis is great, but I wish in my course we'd focussed more on how to prove inequalities, because it's involved in pretty much everything. For your |x|/1+x^2 question, you need to find a another function that it's always less than, and that you know is always less than 0.5.
As for advice on proofs, don't think that proving stuff is the same as the incomprehensible proofs your teachers probably skimmed through when you were younger. It's really just explaining your logic, applying the definitions and theorems you were recently taught in the class, so don't overthink it!
Carl-E:
Wow, a lot's gone by this week. Sorry I didn't get here sooner...
One of the simplest forms of proof is the reductio ad absurdum, or proof by contradiction. For example, in
you can try assuming that it's less than .5. But that means 1 + x2 < 2|x|, and so 1 - 2|x| + x2 < 0. But that means that (1 - |x|)2 < 0, which is a contradiction to x being real (real squares aren't negative).
Sometimes you just need to play with it for a bit.
The proof that 2 is the min for set A is similar, multiply through by x (since it's positive) and you get x2 + 1 > 2x, subtract the 2x and you get a perfect square that's positive. So if you assume x + 1/x can be less than 2, you get the absurdum that the square is negative.
Really, you can prove a lot of stuff by saying "suppose not", and then showing that there's a problem!
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