Is Transition to Advanced Mathematics Too Hard for Me?

[Pengwuino]

Ok I am enrolled in this Math class called "Transition to advanced mathematics". It has basically a LOT of proofing and set theory and such. Now of course, since i went through the whole modern day education system where you memorize this, memorize that, don't question this for the AP test... I feel like I walked into an ambush and I'm going to be destroyed by forces far greater then me.

My question. I passed 1st and 2nd semester calculus and I am currently... uhm... re-taking 3rd semester :) Does this other Math class seem like too much work for me? I am currently taking 18 units with 2 pathetic classes and 2 hard classes. The course description is as followed..

Prerequisite: MATH 76. Introduction to the language and problems of mathematics. Topics include set theory, symbolic logic, types of proofs, and mathematical induction. Special emphasis is given to improving the student's ability to construct, explain, and justify mathematical arguments.

I'm going to ask one of my professors about my situation as soon as possible. That pre-requisite is 2nd semester calculus.

 

[lurflurf]

Such courses are usually pretty light on material. Though they can be of use to the many students who have not picked up the basic ideas of proof and logical thinking that they should have in ealier courses. As far as the prerequisite such courses often require a year or so of college math, but it is not extensively used the "purpose" of the requirement is that the student has some mathematical maturity and so that examples from calculus can be used freely. The ideas introduced such as set theory, logic, and methods of proof are very basic and mostly known to the student, although the symbols and greater emphasis on precision can be startling. Such a course is a good place to find out if one likes mathematics or has any aptitude for it. Start by internalizing all the material. Memorize all of the definitions and symbols. Practice using the concepts and language to express and prove ideas from elementary mathematics and everyday life. Practice the simple proofs that students first learn. Things like
1) There exist an infinitude of primes
2) Square roots of natural numbers are either natural numbers or irrational numbers in particular sqrt(2) is irrational.
3) e is irrational
4) The irrational numbers are not countable
5) The algebraic numbers are countable
6) the fundamental theorem of arithmitic
7) if f'(x) exist and is equal to 0 for all x, f(x) is a constant function
8) if x is nonnegative and x is less than all positive numbers x=0
9) The sum of the first n natural numbers is n(n+1)/2 and similar
10) The harmonic series diverges
Such a course would also be a good place to introduce simple algebraic stuctures such as groups and fields, but although this is often done in textbooks for such classes my understanding is that it is rarely a major topic in courses. Mastery of the basic ideas in a course such as this are important in the transition from computational math courses like that learned in calculus and before, to more theoretical courses taken in the secound year and beyond like algebra, the secound round of calculus, and others.

 

[Pengwuino]

Well the problem is that I have almost non-existant background when it comes to proofs. Only 1 or 2 math teachers back in high school even did proofs for anything and i ended up not having them for any classes.

 

[lurflurf]

Well the problem is that I have almost non-existant background when it comes to proofs. Only 1 or 2 math teachers back in high school even did proofs for anything and i ended up not having them for any classes.

Not even in calculus and trig!
What about geometry no proofs there!
That is why I said you should practice proving basic facts you know. In addition to your assigned text (which is what?) you might consider reading through some elementary books on geometry, calculus, or trig and trying to understand the proofs. Understanding math and understanding proofs are closely related. If you have understood your math so far you should be able to pick up proofs quickly (though with some effort required). If you did not understand your previous courses you may have bigger problems. Can you prove sqrt(2) is irrational for example?

 

[Pengwuino]

Not even in calculus and trig!
What about geometry no proofs there!
That is why I said you should practice proving basic facts you know. In addition to your assigned text (which is what?) you might consider reading through some elementary books on geometry, calculus, or trig and trying to understand the proofs. Understanding math and understanding proofs are closely related. If you have understood your math so far you should be able to pick up proofs quickly (though with some effort required). If you did not understand your previous courses you may have bigger problems. Can you prove sqrt(2) is irrational for example?

I don't even know where to begin on how to prove basic facts

The assigned text is "Chapter Zero: Fundamental Notions of Abstract Mathematics" by Carol Schumacher. I don't ever remember doing proofs as an actual assignment. I don't have time to read through the books because I need to decide if I should keep this course or drop it by Friday. The problem is that the previous courses were easy because it was all formulas and such. I mean you practically had to go out of your way to figure out what exactly you were solving for.

To me its like... being asked to shove my finger in a fan adn then solve the final velocity of my finger using the ideal gas law or use a diarama to solve for y=x^2. I mean i literally have no idea at all how to start proofs.

 

[Brad Barker]

yeah, during the first half of the summer, i took two math courses: complex variables and "numbers and polynomials."

complex variables was cake. algebra and calculus with complex numbers/functions of complex variables. cauchy-reimann equations. cauchy's several theorems, etc. a couple easy proofs for homework, none on the exams. all in all, pretty easy 4000-level (senior-ish level) math class.


numbers and polynomials was an introductory proof writing class. no prior knowledge of proofs assumed. the topics we covered were the algebraic properties of real numbers, natural numbers and induction, elementary number theory, and rational and irrational numbers.

this class was TOUGH for everyone in it! sure, the first chapter was pretty fun--i got to prove stuff like any real number times zero is zero, solely on the axioms that define a field.

there were tough proofs in that chapter, sure, but it wasn't so bad.

but all the stuff after that... ouch. it was VERY difficult to get a hang of proof by contradiction (which you use extensively when dealing with natural numbers).

there was a proof, and you wanted to prove that the antithesis of that was false. but in the process, you would need to prove another thing or two, also by way of contradiction, and so on and so on.



but... with hard work and careful note-taking, i ultimately got it and walked out of the class with the top grade (as far as i know).



your first proof-writing class will be difficult because you are not only USING a completely different skill, but you are DEVELOPING it as well!

fortunately, the professor should be well aware of this and help guide the transition. but like any sort of "birthing" process, it will be very painful at times, but it is essential to experience those difficulties and pains. ultimately, you will grow as ...i'd like to say "person," but in reality, just as a math student.





anyway, i hope that helps.



here are some exercises to get you started:

first, look up "field axioms." only using the field axioms and the properties of a binary operation--look that up, too, i guess--attempt the following:

If a+c = b+c, then a = b.

If a*c=b*c and c is not 0, then a = b.

a*0 = 0 = 0*a

(-1)*a = -a


[and justify each step, too!]

 

[lurflurf]

I don't have time to read through the books because I need to decide if I should keep this course or drop it by Friday.

Well that is not much time to decide. If you can push the class back one term without much disruption to your plans it might be worth it to brush up a bit and attack the course next term with more confidence. Most students have some familarity with proofs from previous courses that is helpful even if they are not well versed. What is more important than background is your confidence in your ability to learn, and your work ethic. That course is one where many students who survived calculus find they are ready to stop.

 

[jmcgraw]

I don't even know where to begin on how to prove basic facts

Penguwuino, get this book:

It is a bit expensive for a paperback, but you can always buy it used. I am self taught in mathematical proof and it helped me a lot. It assumes no proof knowledge, and I think it would suit you well.

Oh, you might also want to go back to your calculus text and look at all the problem, especially from chapters you know well. There are sure to be proof problems that your teacher(s), somewhat foolishly, decided not to assign.

Good Luck!