Apostol 1.19 - Understanding where my logic went wrong (Sets, sup, inf)

[sendthis]

Okay, so I'm struggling with understanding where I went wrong. The instructor feels like I don't understand the material and when she presented my explanation to a colleague, he too agreed with her.

I would really appreciate if someone could tell me the first part of where I went wrong in my logic. If not, what do I need to re-word or what information do I need to provide?

This is question 1.19 in Apostol. Feel free to be as ruthless as you want. I really do WANT to understand and therefore, if I need a reality check, then I'm willing to have my feelings hurt to pass the test. I'm a graduate engineering student that accidentally picked the wrong math class to fullfill my requirement. I have to admit, I do appreciate what the class is teaching but I am having a hard time at getting it..

19) Find the sup and inf of each of the following sets of real numbers.
c) S = {x: (x-a)(x-b)(x-c)(x-d) < 0} where a < b < c < d

I proved that
x > a
x < d

Therefore, I stated since :

The set is bounded above by d, then sup(S) = d (no upperbound greater than d).
The set is bounded below by a, then inf(S) = a (no lowerbound point less than a).

This was accepted by my professor (so I'm aiming that you can accept this without me uploading my proof).

She then asked, "What about a < x < b".

I said that it was irrelevant because for all x, no matter what a, b, c, d (so long as they meet the given problem statement), the set is bounded above and below.

I also said that because any value between a < x < b cannot exist, it is not in the set, so it doesn't matter anyway (by definition of the problem statement).

I am told I am wrong by two PhDs, so now, I need to dig deep and try to understand.

Maybe my logic is right, but my notation is wrong? May be my logic is right but my understanding is wrong? My logic is wrong? May be it's some combination of all the above and I'm a lost cause?

Please help!

I've attached the PDF file. Any comments on notation or concepts I'm missing is GREATLY appreciated.

 

[sendthis]

a < x < b, then S < 0 still applies, I did do it backward. Instead it should be...

x - a > 0
x - b < 0
x - c < 0
x - d < 0

So S < 0...

I realize that careless mistakes such as this can lead to even more confusion...

It doesn't change the result (b/c there are an odd number of '-').

My main emphasis I was trying to get across though, was if

S is bounded above by d (x <d) and there are no points greater than d, it meets the definition of a sup(S) = d.
S is bounded below by a (x >a) and there are no points greater than a, it meets the definition of an inf(S) = a.

Perhaps I got the definition of inf/sup wrong or I am misinterpreting or misapplying it.

 

[willem2]

You've established that d is an upper bound, but not that it is the least upper bound. You have to prove that numbers smaller than d can't be an upper bound.

 

[sendthis]

Hmm... maybe that's where the confusion is. In my original proof, I thought I showed that... I'm looking over it again to see if it's clear. Probably not...

I know I for sure showed it in an earlier proof, but that doesn't mean anything if it's not in my submitted homework.. <sigh>

Thanks. :)

 

[lavinia]

i am a little confused and probably do not understand the question. It seems that (x-a)(x-b)(x-c)(x-d) crosses zero at x=a,b,c, or d so it must have negative values arbitrarily close to zero.

 

[sendthis]

Yeah, I just worked it out in my head. What I wrote in the justification is totally wrong and not what I meant to say. That's probably the disconnect and why two PhDs are saying it's wrong.

My hand was not tracking what my head was telling it to write.

Needless to say, it occurred to me where my original problem was. It was not immediately clear in my proof where I accounted for the fact that d was the least upperbound.

I also think my mind was having problems keeping the words right.

LEAST (and not greatest) UPPER BOUND
GREATEST (and not least) LOWER BOUND

So what I was thinking made sense (to me), I was ineffective in conveying the information.

Thanks for the help. I think it just took me to jam my head a few times to get it.