How difficult is Discrete Mathematics?

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534n
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How difficult is Discrete Mathematics?

Postby 534n » Thu Jul 07, 2011 11:27 pm UTC

So I have placed out of Calculus I, III and Linear Algebra with high school courses, and there is a very good chance Calculus II as well (weird high school set up). I have calculus II in my schedule as a place holder, and the only other available math course currently available to me is Discrete Mathematics, which I would normally take in my junior year for my degree. Could anyone give me insight on the course load or difficulty of Discrete Mathematics (A 300-level course at binghamton, where I will be attending) and whether or not it is wise for a freshman to take? I tried asking my course adviser, but she is set on me taking Calculus II over again even though the mathematics department has told me I can petition to receive credit.

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Re: How difficult is Discrete Mathematics?

Postby Tirian » Fri Jul 08, 2011 2:07 am UTC

Just to save some Google-fu, here is the syllabus at least at some point in the past. (BTW, really Binghamton? 314 is a healthy piece of real estate in the math department and you give it to a discrete course?)

It doesn't look like a killer course -- generating polynomials are a bit of a challenge but the rest of it are a nice exposure to foundational math that I took my freshman year and appreciated having. I'm getting a vibe that they're not expecting heavy proofs from you here which you'd probably appreciate as well. Long story short, if they are actually offering you the option then they probably expect that it is something that a motivated incoming freshman can handle.

Just to play devil's advocate for a moment, I "retook" Calc 2 my first semester in college in a story similar to yours, and I got things out of it (not the least of which being a good tutoring gig). I'm not a fan of cakewalk courses in general, but making sure you're grounded in calc and having at least one oasis in your first semester are advantages worth considering.

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Re: How difficult is Discrete Mathematics?

Postby ++$_ » Fri Jul 08, 2011 4:15 am UTC

We deserve 314 if you ask me.

Anyway, this class looks pretty straightforward. It looks like it should be accessible to first-year undergrads.

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Re: How difficult is Discrete Mathematics?

Postby the tree » Fri Jul 08, 2011 9:20 am UTC

Most of that looks kind of well, essential for any course really: learning how to do proofs in discrete is a pretty good stepping stone to being able to do proofs in real analysis. You need set theory basically all the time (there is not actually such thing as set theory, there's just a lot of terminology to learn - but it's all very useful), and as for the combinatronics at the end, that comes up everywhere.

In short, you will find that course connects directly with whatever maths courses you are taking along side it. Parts of it should be difficult, but that is purely because a maths course is not supposed to be easy.

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Re: How difficult is Discrete Mathematics?

Postby Yakk » Fri Jul 08, 2011 2:00 pm UTC

For a "theoretical/pure" math student, much of that is material I'd be disappointed they didn't cover until 3rd year.

Line by line:
Spoiler:
Chapter 1: Yes, There are Proofs!

Compound Statements
Proofs in Mathematics

Needs to be covered, but by 3rd year, haven't you done proofs before?
Truth Tables

Why would this be worth mentioning as a subject covered in a math major course?
The Algebra of Propositions
Logical Arguments

Interesting material could hide behind these subject descriptions.
Chapter 2: Sets and Relations

Sets
Operations on Sets
Binary Relations
Equivalence Relations
Partial Orders

Could be covered by a 1st year student, in my experience.
Chapter 3: Functions

Domain, Range, One-to-One, Onto
Inverses and Composition
One-to-One Correspondence and the Cardinality of a Set

Could be covered by a 1st year student, in my experience.
Chapter 4: The Integers

The Division Algorithm
Divisibility and the Euclidean Algorithm
Prime Numbers
Congruence
Applications of Congruence

I could see delaying this until later years: my alma mater covered it in first year, but it was a quirky choice.

I'd expect the Chinese Remainder Theorem to be here, actually. And formal polynomials, with basic ring extensions. But I guess they tossed a bunch of combinatorics into the course.
Chapter 5: Induction and Recursion

Mathematical Induction

Wait, 3rd year before you cover induction?! How did you prove ... well, much of anything ... before 3rd year?
Recursively Defined Sequences
Solving Recurrence Relations: The Characteristic Polynomial
Solving Recurrence Relations: Generating Functions

These are fun topics, and worth a few weeks.
Chapter 6: Principles of Counting

The Principle of Inclusion-Exclusion
The Addition and Multiplication Rules
The Pigeon-Hole Principle

1 cookie, bwahahahahah
2 cookies, bwahahahaha
Chapter 7: Permutations and Combinations

Permutations
Combinations
Repetitions

I choose you, combinatorics.

What is the intended audience of this course -- all computer science and mathematics students, regardless of discipline? What is your intended specialization?
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Re: How difficult is Discrete Mathematics?

Postby Tirian » Fri Jul 08, 2011 4:25 pm UTC

Yakk wrote:Needs to be covered, but by 3rd year, haven't you done proofs before?


The closest the standard American comes is about a month worth of "Statement/Reason" tables in plane geometry. Writing out solid arguments in paragraph form would start here, probably even for college juniors at second-tier math schools. :(

Truth Tables

Why would this be worth mentioning as a subject covered in a math major course?


IME, for now they'll define tautology and contradiction in terms of the results of the truth table. I trust they'll spend an evening using the tables to verify the DeMorgan laws, frex. Things get more axiomatic in a first-order logic class, but that's probably put off for grad school.

The Algebra of Propositions
Logical Arguments

Interesting material could hide behind these subject descriptions.


Yeah, this is arguably the most applicable part of the course. Understanding direct proofs, contradiction, contraposition, proofs by cases, and knowing when to use them in specific cases is so critical not just to math but all of higher critical thought. Once you're through breaking into this junior course, you should figure out Binghamton's version of the Problem Solving Seminar course is and get yourself into that next so you can actually apply these concepts in a wide variety of contexts. Like linear algebra, appreciating proofs is one of those things that increases understanding even in low-level courses for the students who come in with those talents.

I'd expect the Chinese Remainder Theorem to be here, actually. And formal polynomials, with basic ring extensions. But I guess they tossed a bunch of combinatorics into the course.


This is the first formal exposure to discrete math. They haven't had abstract group theory, much less rings. I'm inclined to agree about CRT, although I confess it didn't come up in my discrete course either (or anywhere else in Carnegie Mellon's program, to be honest :oops:).

Wait, 3rd year before you cover induction?! How did you prove ... well, much of anything ... before 3rd year?


Heh. Students are asked to follow proofs in introductory continuous math (if they choose to not trust the textbook and professor), not construct them. I hope most students have already had induction, but with 50 different state curricula and all the international students American colleges probably think it's safest to spend a day on it when they'll need it to justify the definition of recursion.

Recursively Defined Sequences
Solving Recurrence Relations: The Characteristic Polynomial
Solving Recurrence Relations: Generating Functions

These are fun topics, and worth a few weeks.
[...]
I choose you, combinatorics.


Agreed. I'm glad I got a whole course of combo instead of just a third of a semester in a survey course like this. It's worth it. The one bright spot is that they're evidently saving graph theory for its own course.

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Re: How difficult is Discrete Mathematics?

Postby AMathematician » Fri Jul 08, 2011 4:37 pm UTC

I'm at a small liberal arts school going for a math degree, and I saw proofs my freshman year. I was a little ahead of the curve with some AP credits, but you would almost certainly see something beyond the crappy geometry tables you saw in high school before your junior year. My first experience was with linear algebra, so if your school focuses less on proofs for that class YMMV.

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Re: How difficult is Discrete Mathematics?

Postby Hecate » Fri Jul 08, 2011 8:33 pm UTC

I'm pretty sure this syllabus is exactly what was in the (Second year) introductory Discrete Math course I took in my Freshman year of university. I personally found that course to be really easy, and I hadn't done enough math in high school to get any credits. If you tested out of Calculus I, II, III, and Linear Algebra, I can't imagine this course being very hard for you. A number of people in my class had a hard time with it, but I have a feeling that was because we had a pretty mediocre prof, and because a lot of the students didn't really try. If you study the notes, you'll be fine.

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Re: How difficult is Discrete Mathematics?

Postby Tirian » Fri Jul 08, 2011 9:10 pm UTC

++$_ wrote:We deserve 314 if you ask me.


Granted, discrete is a critically important course, I'm just thinking it's one of the few courses in the undergraduate curriculum that doesn't give a flip about pi. Perhaps 358 would be more appropriate as a subsection of the fibonacci sequence. Pity it isn't a sophomore course, because 293 would be an even more awesome choice as the answer to the seminal question used to introduce generating polynomials.

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Re: How difficult is Discrete Mathematics?

Postby doogly » Fri Jul 08, 2011 9:33 pm UTC

Take calc 3 and discrete math. Spending the first week or two of class in calc 3 will help make it quite clear whether you should be there or in calc 2.
And you should certainly try to get credit for calc 2. That's why they make these placement exams.
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Re: How difficult is Discrete Mathematics?

Postby themandotcom » Sun Jul 10, 2011 10:31 pm UTC

I actually go to binghamton!

discrete wasn't too bad, but i found calc 3 mad hard.
e^pi*i=WHAT??

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Re: How difficult is Discrete Mathematics?

Postby Nat » Tue Jul 12, 2011 9:07 pm UTC

That looks easier than our first year discrete math course, which isn't that hard. so yeah, do it as a freshman. (seriously, how can you possibly have math degrees with no proofs until third year? it's like having a literature degree where you don't read any books until third year).

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Re: How difficult is Discrete Mathematics?

Postby Dason » Wed Jul 13, 2011 4:21 pm UTC

Well I find it hard to believe that you could get the necessary prereqs to read books before the third year. You need to do all the work to prove the existence of books in a constructive manner. Then you need to prove Jensen's Page Turning theorem (or else you'll never get anywhere)! Maybe you could squeeze most of that into the first year and start with books by the second year but that seems pretty intense to me.
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Re: How difficult is Discrete Mathematics?

Postby freakish777 » Wed Jul 13, 2011 7:03 pm UTC

You should try and place out of Calc II and take the Discrete Math course.

The less classes you're "required" to take, the better:

A) You can potentially finish early and skip out on a semester or two of tuition. Even if it's only $3,500 a semester that's $3,500 less in loans. More importantly, that's 4~6 months of work (or collecting unemployment, or grad school) you get to do instead of being in school.

B) Don't want to graduate early? More time for electives. Electives are great for actually deciding what you want to do, and exploring your options (I took an Astronomy & Astrophysics course as an elective. Turns out I don't like that field at all, but if I had liked the course, maybe I would have applied for some of the research internships with the professor of that class and been doing something entirely different right now). They're also great for doing Master's level classes as an underclassman (provided your school let's it's Senior Undergrad take Master's level courses).

C) More time for Co-Ops, Internships, Undergrad Research Projects.


Basically, you should try and skip out of every single required course you can, since the pay off for the above 3 things tends to be greater than the pay off of learning the few things in the required class you didn't already know.


Also, Discrete Math is simple (given the right professor).

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Re: How difficult is Discrete Mathematics?

Postby amit28it » Tue Dec 13, 2011 1:01 pm UTC

Can you help me with this logic problem for my discrete math class?
Given the facts below you must decide, if you can, who committed the crime. You must use logic to solve the problem. Translate the facts below into logic propositions and use a truth table and or logical equivalences to arrive at a conclusion. If you can determine who committed the crime, explain how the logic brought you to this conclusion. If you cannot determine the criminal, explain why using logic.



(a) If A is guilty and B is innocent, then C is guilty.

(b) C never works alone.

(c) A never works with C

(d) No one other A,B, or C was involved, and at least one of them is guilty.

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Re: How difficult is Discrete Mathematics?

Postby doogly » Tue Dec 13, 2011 1:33 pm UTC

propositions b and c conspire to give you a good dose of information about suspect C.
So now, look at your options. There are 6 combinations you can be considering. First ask, is C guilty or innocent?
Either is possible just given b and c.
So now look at the rest. If C is guilty, which combinations of A and B innocent/guilty are possible, given all the propositions?
Which combinations of A and B innocent/guilty are possible, if C is innocent?

After accounting for all the propositions, are any combinations left? Are more than one?
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Re: How difficult is Discrete Mathematics?

Postby Proginoskes » Wed Dec 14, 2011 8:17 am UTC

That's a problem from one of Smullyan's books. (One of the easy problems, I might add ...)

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Re: How difficult is Discrete Mathematics?

Postby amit28it » Wed Dec 14, 2011 11:21 am UTC

Discrete Math question regarding combination equations?
so there are two equations for combinations with repetitions:
one is:
C(n+r-1,r)
and the other is:
C(n+r-1,n-1)

ive seen separate examples using both but they can never be overlapped/switched in those cases, when do i use which equation then? which one is universal? how can i tell when i should use them?

thanks for your time.

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Re: How difficult is Discrete Mathematics?

Postby jaap » Wed Dec 14, 2011 4:29 pm UTC

amit28it wrote:one is:
C(n+r-1,r)
and the other is:
C(n+r-1,n-1)

They're equal - Pascal's triangle is symmetric.

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Re: How difficult is Discrete Mathematics?

Postby dissonant » Fri Dec 16, 2011 4:34 pm UTC

For what it is worth, my favourite course as an undergraduate was discrete mathematics. I feel that, often, the abstract nature of mathematical reasoning is disconcordant with the natural world. However, I never feel this way with discrete mathematics. It seems so emminently applicable, and indeed, every time I recall myself needing a mathematical problem solved in the real world, the majority were discrete.

For a slightly more controversial stance on the position, I recommend the following article by Doron Zeilberger, "Real Analysis is a Degenerate Case of Discrete Analysis." http://users.uoa.gr/~apgiannop/zeilberger.pdf

Although, if you are convinced by his words you may wish to watch his lectures. http://www.youtube.com/watch?v=ZpI1BWr_Ixo ;)

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Re: How difficult is Discrete Mathematics?

Postby doogly » Fri Dec 16, 2011 4:48 pm UTC

This was controversial when Kronecker was talking about a watered down version. By now it is just ridiculous.
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Re: How difficult is Discrete Mathematics?

Postby Marbas » Sat Dec 17, 2011 11:59 pm UTC

doogly wrote:This was controversial when Kronecker was talking about a watered down version. By now it is just ridiculous.


I like how his philosophy basically throws out topology as a legitimate branch of mathematics.
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Re: How difficult is Discrete Mathematics?

Postby jestingrabbit » Sun Dec 18, 2011 12:35 am UTC

No, no, it just relegates it to a "degenerate approximation... to the discrete world, made necessary by the very limited resources of the human intellect."
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Re: How difficult is Discrete Mathematics?

Postby doogly » Sun Dec 18, 2011 4:36 am UTC

I wonder if he thinks Wolfram is a decent scientist, too. It all seems kind of adorable.
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