[MATH] Proofs / "logic"

[AwesomeMcCoolName]

How would I prove what inf(s) and sup(s) is? 

 

S={x: x^2<5x} 

Obviously the bound is (0,5), but how would I prove inf(s)=0, and sup(s)=5? 

 

S={x: 2x^2 < x^3+x}

The bound is is (0,1)U(1,infinity), so how would I prove inf(s)=0, and sup(s)=DNE? 

[Bonzo The All-Gnawing]

You don't.

[fluffyy]

2+2

[Pwndaz!]

I was expecting basic geometry in this thread... boy oh boy was I wrong.

[Mole]

as a dutch mathematically interested person, this doesn't make sense to me

[Liddojunior]

I'm only starting infin series this week can't help you buddy. 

 

And most of these responses are beyond not helpful

[Karam]

Math has logic ?

[AwesomeMcCoolName]

I'm only starting infin series this week can't help you buddy. 

 

And most of these responses are beyond not helpful

This isn't calculus  

[The Phantom]

I'd be glad to help you!  

 

Couple things:

1. In general, math text doesn't translate to english text.  Are there any symbols or otherwise that didn't type well?

2. What subject is this, it will help me narrow it down.

3. What do inf(s) and sup(s) stand for?

[Liddojunior]

This isn't calculus

 

I know but you still need to know infinite series. It was more of a nod, that you're ahead. 

[AwesomeMcCoolName]

I'd be glad to help you!  

 

Couple things:

1. In general, math text doesn't translate to english text.  Are there any symbols or otherwise that didn't type well?

2. What subject is this, it will help me narrow it down.

3. What do inf(s) and sup(s) stand for?

1. No

2. Real Numbers / Analysis I think 

3. Infemum and Supremum, greatest lower bound and least upper bound respectively. These are all pretty universal terms, so if you don't recognize them, you probably won't be very helpful I'm afraid  

 

 

I know but you still need to know infinite series. It was more of a nod, that you're ahead. 

You actually don't  

[Python.]

I would know if I paid attention in class

[Toastshpee]

I was expecting basic geometry in this thread... boy oh boy was I wrong.

So was I.

Geometric proofs suck.

[The Phantom]

1. No

2. Real Numbers / Analysis I think 

3. Infemum and Supremum, greatest lower bound and least upper bound respectively. These are all pretty universal terms, so if you don't recognize them, you probably won't be very helpful I'm afraid  

 

 

You actually don't

 

"Pretty universal terms"... weird, because after having taken Calc I through III, differential equations, linear algebra, prob and stats, signals and systems feedback control and design, etc etc etc, I've never heard of them.  

 

How do you need these proven, proof by induction or otherwise?

[The Phantom]

Actually, I'll just assume this doesn't require proof by induction, it doesn't seem like that sort of problem.  Correct me if I'm wrong.

 

When dealing with any values approaching infinity, it's best to not actually use the number infinity.  The most mathematically (and logically) correct way is to write as lim as x-->inf, or "as x approaches infinity.  We can't make any assumptions here, because quite frankly infinity behaves very oddly.  Zero on the other hand, is easy (unless combined with inf again).

 

To be specific, for problem #1, write as such:

S=lim x-->0 x^2<5x

Show that for x<1, the x^2 value becomes smaller than x.  This implies that for all 0<x<1, x is of the set S.  Thus we're approaching your lower bound.  Now actually plug in 0 to confirm that x^2 = 5x, which is not within your set.  From these two statements, you were able to prove as x approaches zero, x is within the set, but zero is not a part of the set.  inf(s) proven.

 

 S=lim x-->5" x^2<5x

Now we will actually re-write this as:

 S=lim x-->5" x*x<5x

This does some mathmagic for us, because it allows for a direct comparison between 5 and x.  Since both sides are multiplied by x, we can wipe it out.  

S=lim x-->5" x<5

And thus, this part has nearly proven itself.

 

For #2:

S=lim x-->0 2x^2 < x^3+x

Start by dividing by one power of x on both sides.  This yields:

S=lim x-->0 2x < x^2+1

From this, show that when x<1, x is of the set S.  Like above, when it reaches zero, both sies are equal (don't forget there's still a factor of x on the outside, not written here).  Then x=0 is not of the set s.  In doing so, you have also proven that x=1 is a disjoint in the set.  

S=lim x-->inf 2x^2 < x^3+x

Here comes the tricky one... infinity.  You've gotta be careful here.  I could go through a method to explain how you'd do it for infinite series in calc II, but I get the impression it's beyond the scope of the course.  Basically one way you could do it is with a replacement of variables.  Check this out:

S=lim n-->inf 2n^2 <n^3+n

S=lim n-->inf 2(n+1)^2 < (n+1)^3+(n+1)

Start by setting n=2 or something (simple, yet above 1).  you get 8<9.  Works out.  Now we expand the n+1 expression.  

S=lim n-->inf 2(n+1)(n+1)<(n+1)(n+1)(n+1)+(n+1)

S=lim n-->inf 2(n^2+2n+1)<(n^3+3n^2+3n+1)+(n+1)

We can see that at least n^2+2n is present in the second half, but larger as 3n^2+3n.  Then there's still the n^3 expression and more.  If this isn't sufficient, one could argue that as n-->inf, each n^x goes to inf.  then you get:

S=lim n-->inf 2(inf+2inf+inf)<(inf+3inf+3inf+1)+inf

S=lim n-->inf 8inf<9inf

Since n is true, we can argue n+1 is true.  Then, any number 'n' plus 1 is true.  This creates a series of integers, where each is proven from the previous, up to infinity.  

 

Hope your HW wasn't already due.

P.S., I know them as GLB and LUB.

P.P.S. This may be a bit lengthy, it might be in your favor to pick and chose which steps show the most work without including every tiny detail.

[>Sneaks]

I have a feeling this is more advanced than Calc II. Doesn't look like anthing I've ever come across. Are you an engineering major?

[AwesomeMcCoolName]

Actually, I'll just assume this doesn't require proof by induction, it doesn't seem like that sort of problem.  Correct me if I'm wrong.

 

-snip-

 

Hope your HW wasn't already due.

P.S., I know them as GLB and LUB.

P.P.S. This may be a bit lengthy, it might be in your favor to pick and chose which steps show the most work without including every tiny detail.

lol, stop making sense. I'm supposed to prove this shit with theoretical bullshit logic.

 

Basically something like this.. 

S clearly has a lowerbound at 0 and an upperbound at 1. 

Assume u=5 is an upperbound of S, if u<=M' where M' is another uppbound of S, then u is the LUB of S. 

(or, more likely, Assume u<5, then by density of irrational numbers, there exists an I such that u<I<5. Therefore 5 is the LUB).

I have a feeling this is more advanced than Calc II. Doesn't look like anthing I've ever come across. Are you an engineering major?

yes

[The Phantom]

Not sure if he was asking you or me about being an engineering major... but yes.

 

"Theoretical bullshit"... is this a discrete mathematics class? 

 

If it's Calc II, there's no theoretical anything.  As my professor said to us specifically about inf. series "No handwaving bullshit."

[AwesomeMcCoolName]

Not sure if he was asking you or me about being an engineering major... but yes.

 

"Theoretical bullshit"... is this a discrete mathematics class? 

 

If it's Calc II, there's no theoretical anything.  As my professor said to us specifically about inf. series "No handwaving bullshit."

It's fundamental mathematics; so ya, basically discrete math.