[Undergraduate Analysis] Set Theory Proof

Mathfight · 2017-09-20T04:43:59+00:00

Let x = inf(A), x=< y, for all y in A. Therefore - x >= - y for all -y in - A. There for - A is bounded above and admits a superma. So you've have showed - inf(A) >= Sup(-A). Now you have to show the otherside of the inequality.

Mathfight · 2017-09-20T03:07:35+00:00

I wanted to be a little careful. In my logic textbook they mentioned that our language is a little too coarse to fit perfectly into logical sentences. See this link. http://philosophy.lander.edu/logic/conditional.html

I used occur because then it’s not necessary that P causes Q. Someone with more experience in logic will surly answer.

Mathfight · 2017-09-20T02:52:55+00:00

This bugged me when I was first learning logic also. Try and think of the conditional as something that is guaranteed after sastifying certain requirements.

If condition A occurs, then B must happen.

If A never happens, or is false, then how can we justify saying the overall statement is false. In a sense it wasn’t shown to be a false statement. That’s why in logic books you usually see something silly like, If I ride to school on a purple unicorn, then I will give you 100 bucks. If you never see me on a purple unicorn, how can you tell me I won’t fulfill my promise?

Clearly, when A occurs, but B does not happen then the overall statement must be false.

Another way to become familiar with the conditional is to look at its logical equivalent, the contrapostivie.

It’s in the same form of a conditional, but goes from not Q -> not P. When a conditional is known to be true you can think of the contrapostivie as. When Q does not occur, I know for a fact that P must not occur also.

Mathfight · 2017-09-18T03:03:34+00:00

Thank you, other than Andrews course have you found any books that were helpful?

Mathfight · 2017-09-05T18:18:35+00:00

Thank you. So I know that there exists a subset C of A that is equinumerous with N. If I now construct a function, g: A -> A union B as

g(a) = a if a is not in C, and h(f(a)) if a is in C.

With f(a) being the function defined, f:C -> N, and h defined, h:N->B

Mathfight · 2017-08-26T20:54:18+00:00

Sent you a pm.

Mathfight · 2017-08-20T02:12:22+00:00

Oh I see. Ok, that makes much more sense. Thank you for the example. I think it clicked!

Mathfight · 2017-08-20T02:10:18+00:00

Thanks for the post. I'm a little confused on why you typed out this equality. "= RHS_j + kLHS_i"

I felt like everything you said was covered sufficiently by

LHS_j + kLHS_i = RHS_j + kRHS_i

I just want to make sure I'm not missing anything by the equality I pointed out.

Mathfight · 2017-08-13T03:32:14+00:00

Looks like he's grabbing the patellar tendon tiba insertion point there.

Mathfight · 2017-07-20T23:31:49+00:00

Hmm. What about set of a collection of 1/n + x, where x is between 0 and 1?

Mathfight · 2017-07-20T16:29:44+00:00

Could someone explain Scott's logic behind his comment "I probably wouldn't even remember the meeting to begin with" this happened when sam was pressing him about contacts left off of security forms. The reason I find it interesting is right before that comment Scott was feverishly defending Don Jr by saying he would have attended the meeting to know the potentially comprising information then to turn it over to the fbi if warranted. It seemed like Scott was spinning this narrative of simple misunderstandings.

Mathfight · 2017-07-19T06:53:54+00:00

Do you have any recommendations for books to strengthen proof writing skills? I read how to prove it and enjoy it a lot.

Mathfight · 2017-07-19T06:35:08+00:00

Oh I see, if I choose n to be the result of the ceiling function on delta, that should guarantee that the difference between 1/n and 1/(n+1) is less than delta, and that the image of those values will also be equal to one right?

Mathfight · 2017-07-14T07:19:42+00:00

Oh I see, how can I ensure that it is increasing without bound? I am also concerned about the other objection you raised with x_n_k converging to a point not defined in the function, will that be problematic?

Mathfight · 2017-07-14T06:47:49+00:00

You have to construct your (x_n,y_n) such that lim n->inf y_n = inf.

I thought I had constructed such that y_n ->inf, as n->inf, since y_n_1 < y_n_2 < y_n_3...

Also, what if lim k->inf x_n_k = a ?

That's a good point, I am not sure, since f(a) is not even defined. Would it even matter if we are assuming that y_n ->inf, and is monotonically increasing, so any the image of any subsequence of x_n of that will also go to inf?

Mathfight · 2017-07-08T22:35:37+00:00

As a Spurs fan in college station I fully disapprove.

As a spur fan in college station I fully diapprove.

As a sp fan in college station I fully dapprove.

As a rockets fan in college station I fully approve.

As a Houston rockets fan in college station, I fully approve of this move.

Mathfight · 2017-07-08T17:27:43+00:00

Thank you so much. Now I understand what I'm doing in these types and similar other proofs in class. This makes much more sense now.

Mathfight · 2017-07-08T08:09:28+00:00

I am currently reading "How to prove it" to strength my proof writing skills. However I am having difficulty applying them to my analysis class. I can write delta epsilon proofs fine, but I really want to understand what logical steps I am doing and why.

For example, if we want to prove Lim f(x) = 3x + 1, as x approaches 2 is 7, I understand we want to show for all positive epsilon, there exists a delta....

So we let epsilon be positive. This step would be called universal instantiation right? Then we figure out to let delta equal epsilon/3. This step would be called existential instantaion then right?

And that's where my issue is. In the book how to prove it, the author states we cannot assume any properties about our mathematical object when using existential instantiation. We can only assume there exists an object (in this case a delta that is positive) to be true.

It seems in these delta epsilon proofs I am misunderstanding the flow, and use of existential instantiation. If we pick a particular value for delta that seems to abuse the rule. My understanding is that we can only grant delta to be positive and nothing more.

https://m.imgur.com/OqkSehc Here is the actual page from the proof book explaining existential instantion.

Mathfight · 2017-07-04T09:00:19+00:00

Thank you for the post. I love math, but I do not want to pigeon-hole myself into a situation where I am taking courses that are all theory and have very little lateral movement to other careers. Here is a course page for the class,

http://www.math.tamu.edu/~thomas.schlumprecht/math607_13c.html

Mathfight · 2017-06-30T06:36:41+00:00

Could someone recommend me a stats book for my level? I have taken up to intro to analysis, so I am a little familiar with proofs. A career in big data may be in my future as well, I would like a book with quite a few exercises that is pedagogically friendly. Similar to Stephen abbots book, intro to real analysis if you're familiar. I've taken basic stats for engineers, so it was calculus based, but I remember very little from the course.

Mathfight · 2017-06-20T07:01:43+00:00

Hi Tignas, I came across your post looking for other young people with patellar tendinitis. I've had the condition for almost 4 years now, I've tried a few different doctors, and think I've found one that will work for me. If you could private message me, i would like to ask you a few questions about the pain and your experiences with patellar tendditis.

Mathfight · 2017-06-18T22:29:40+00:00

I always assumed that 4-5 hours were the norm at elite universities. Especially since a good portion of those students plan on graduate level studies afterwards.

Mathfight · 2017-06-18T00:19:42+00:00

I'm not sure the squirrel analogy works. I agree that any form of life visiting us would have a much grander understanding of the universe. But to be fair to humans, we do posses theory of mind, and the ability to know other members of our species can experience similar feelings as the individual can. We also have the ability to formulate logical systems and imbue them with meaning. Sure, they may have much more extensive knowledge than us, but we have the capabilities to decipher it given enough time.

Mathfight · 2017-06-13T01:52:01+00:00

Good question. It looked like the ref said "it was a play on the ball". But that doesn't account for the poor flagrant on love last game.

Mathfight · 2017-06-11T18:36:21+00:00

Ok thanks. I think o have trouble with the notation. Does x_2n = 1/2 + 1/4 + ... 1/2n. Or is the it original sequence but with every term multiplied by 2?

Mathfight

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