Introductory Real Analysis - check answers please 4 Questions

[playboy]

Hey, university started and in less than a week, we already have to hand in an assignemnt. I did most questions, i was just wondering if somebody can help check some of work - its more english than math

Q1) True or False - A statement and its negation may both be false.
A1) False - if p is the statement and is true, then ~p is false and visa verca. (They HAVE to be opposites, they cannot be both True or both False)


Q2) Write the negation of each statement
...a)the set of rational numbers is bounded
...b)if f is continuotus, then f(S) is closed and bounded
A2)
...a)the set of rational numbers is NOT bounded
...b)if f is continuotus, then f(S) is NOT closed or NOT bounded


Q3) identify the antecedent and the consequent in each statement
...a)a sequence is convergent provided that it is monotone and bounded
...b)convergence is a sufficient condition for boundedness
A3)
...a)antecedent: monotone and bounded... consequence: convergence
...b)antecedent: boundedness... consequence: convergence


Q4) Let p be the statement "Buford got a C on the exam" and let q be the statement " Buford passed the class". express these 2 statements as symbols:
...b)if Buford passed the class, he did not get a C on the exam.
...a)it was necessary for buford to get a C on the exam in order for him to pass the class
A4)
...a) q --> ~ p
...b) no idea

I know the above is kind of long and above, but if somebody can please check it over, id greatly appreciate that. Thanks in advance!

 

[CarlB]

I counted 4 wrong answers. Yes, it is more English than math, but that is what higher mathematics is like a lot of the time. If I post answers I'm just going to get my post erased by the authorities here and it wouldn't help you much anyway. Look over your list and try again.

You may find it useful to pick up a book for cramming the LSATs. The reason is that the use of English in the legal professions is somewhat similar to the use in mathematics. The reason for prepping with the legal stuff is that there's a lot more help books in that area. Nobody wants to be a mathematician, everybody wants to be a lawyer.

Carl

I had no idea that this is how mathematics is being taught today, but I have to say that it is a considerable improvement from 30 years ago. Back then it was clear that a lot of students just weren't picking up the ideas but this sort of first week's assignments makes it very clear exactly what the problem is. You can't imagine how much more difficult the later problems in this class will be, in terms of interpreting the English correctly. They're doing you a big favor by bringing this up so early in the class. When I took classes like this they sucked the students in with easy stuff and then hit them hard by the end.

 

[matt grime]

If X then Y is the same as not(X) OR Y hence its negation is X and not(Y)

 

[Mathechyst]

I had no idea that this is how mathematics is being taught today, but I have to say that it is a considerable improvement from 30 years ago. Back then it was clear that a lot of students just weren't picking up the ideas but this sort of first week's assignments makes it very clear exactly what the problem is. You can't imagine how much more difficult the later problems in this class will be, in terms of interpreting the English correctly. They're doing you a big favor by bringing this up so early in the class. When I took classes like this they sucked the students in with easy stuff and then hit them hard by the end.

It is this fundamental mathematics that I consider to be part of a transition course. Such a course should also include all the nitty gritty about functions, relations, and a few other things as well as how to write a proof. Unfortunately, all too many schools still don't insure students know this stuff before they let them take higher math courses. I was unable to get the hang of higher math until I figured out on my own that this was what I was missing.

Ideally, this should be taught in freshman calculus. Limits would have been so much easier to understand. But, since freshman calculus courses are mostly populated by future engineers and engineers have a propensity to only want to know how to calculate things, perhaps professors don't try anymore. At least, this was my attitude when I was in my engineering phase.

"Analysis with an Introduction to Proof" by Steven Lay is a great book to have whether you know this transition material or not. I studied undergraduate analysis out of Rudin's book and Lay's book was an essential companion.

Doug

 

[matt grime]

But, since freshman calculus courses are mostly populated by future engineers and engineers have a propensity to only want to know how to calculate things, perhaps professors don't try anymore.

I assure you nothign would give us greater pleasure than to teach mathematics properly, however the students don't want to hear it. And students write evaluations of the teachers that are not very kind in general especiallly if the lecturer is giving material the student deems unnecessary (it is not important to them to understand mathematics but to merely be able to compute enough examples to pass the course). If you get bad teaching evaluations you have to explain them, they are on record and they make it harder to get jobs. But that's the system for you: the student knows best, which makes you wonder why they need to be in class.

 

[playboy]

Hmm i agree with you matt grimm.

This is Real Analysis - I am using Steven Lays book for the course.

All i care about is computing equations right now... but i guess this course is more about explaining stuff - and i hate that.

Well.. I am going to re-look over a few questions then ill repost them and please help me out! thanks in advance

 

[playboy]

Okay i have another question for you guys. From the book, the question has backwards E's and upside down A's but I am just going to translate it into english

Q) Which of the following best identifies f as a constant function, where x and y are real numbers. (Their were 4 questions, but i narrowed it down to 2)

A)
a) For every x their exists a y such that f(x) = y
b) Their exists a y such that for every x, f(x) = y

Okay.. these both look completely identical too me. I see no difference at all, just that they are written backwards from each other.

I chose a, know that it is correct, but i just cannot see why b is wrong? anybody have an ideas

Thanks in advance.

 

[Mathechyst]

I hate to be the bearer of bad news but B is the correct answer. Before I explain why you should reread the section on quantifiers and see if you can come up with an explanation. By the way, the ordering of quantifiers is very important.

Doug

 

[matt grime]

what does a) say? well, assuming that we mean to think of f as possibly being a function from the reals to the reals, then it says for every input value there is some output value (note it doesn't say that it is the same output, it is allowed to depend on x), which is just saying that f maps the reals to the reals. remember that when you see a

for all A there is B

that B is not always the same and depends on A (in general)

eg:

for all positive numbers there is a positive square root.

now you don't think all numbers have the same sqaure root do you?

 

[playboy]

oh goodness...

their were examples in the book just like this... but its very hard to see.

so a is wrong because it says "for any x their is (any) y such that f(x) = y" and y can be any number, not a constant.

and b) is the correct answer since it gives exactly 1 value for y if any x is in f(x) = y