I started reading a little bit about basic set theory. A lot of the basic stuff says things like if Set A equals Set B, then any member of Set A is also a member of Set B. A lot of what I've read seems like common sense. Does it go any deeper than this? When you get deeper into set theory, does it get any more useful or insightful? I don't want to waste time with it if it's just a bunch of common sense. I might just learn the symbols so that I can recognize them. Are there any real uses of set theory?
Yes. Set theory appears in quite a different number of places in advanced mathematics (like measure theory), but indeed you need to get familiar with the basics before you can reason in more advanced and abstract settings. Bye! Crisp
set theory is very important. just how important it is, i think you will fail to see until you have a head for abstract mathematics. once you do, you will see that set theory is the foundation on which the house of mathematics is built. the axioms seem common sense to you. in fact, that is something that you will find in any branch of mathematics. the axioms of algebra are a bunch of arithmetic rules that you already know. they are "common sense". same with set theory, topology, and geometry. each subject starts with axioms that almost seem trivial, to the uninitiated. but rest assured, even though the axioms seem intuitive, the mathematics can be very difficult. the payoff is that the results will be quite general. one of the most basic ideas of set theory is the idea of the equivalence relation. a relation R on a set X is a subset of the power set of XxX. xry iff (x,y) is in R. it is an equivalence relation iff (x,x) for all x, (x,y) implies (y,x) in R for all x,y, and finally (x,y), (y,z) in R implies (x,z) in R. set theory shows that every equivalence relation is in one to one relation with partitions of the set (a partition is a collection of subsets that are pairwise disjoint, and whose union is the whole set). now, did you follow all that? it might have seemes difficult. if so, then you need to learn more set theory. it is not as trivial as it looks on first glance.
jwsiii, when I started my Discrete Math course at uni I had exactly the same thoughts. Now, when I begin to work through problems such as probability, logic, and relations, my brain thinks of everything as sets (in some form or another). What I'm trying to say is that Set Theory is very useful when dealing with finite numbers like sample spaces in probability or pigeonhole theory. A shortlist of important stuff that requires an understanding of sets: * Relations | Heard of reflexive, symmetric and transitive? A big section of mathematics depends on the relationship between one set of numbers to another. If a relation is all three then it is equivalent and all of a sudden you unlock a range of wonderful things! Of course all this would not be possible without set theory (How do you think you determine R, S and T in the first place?) * Permutations & Combinations | "How many ways can six oranges and 4 apples be arranged so that no two apples are next to each other?" A common problem that can be related to many other like it. Set theory is the very foundation of permutations and combinations, without those rules that bound set theory together the above problem would be very difficult indeed!! * Probability | Bookmakers beware! If you know set theory then probability becomes as easy as pie. Probability is all about sample spaces and these spaces are very horrible places when they become large. Set theory gives you a nice framework that cuts down the sample space making probability questions a breeze. Also lots and lots of probability theorems are structured by set theory. * Binomial Theory | Pascal's triangle can get very big and nasty after ten or twenty iterations. So how could anyone possibly work out those binomial expansions when you have hundreds of terms? Well, set theory makes this so much easier. As I said before, the workings of set theory breaks it all down into chewable bits. * Recurrence Relations | If only monks knew about set theory when they tried Tower of Hanoi with 60 discs! * Graph Theory | A much more recent topic that is almost completely based on the advantages of set theory. This is very important front of mathematics giving rise to many unsolvable but extraordinarily useful problems. Travelling Salesperson problem to name one. And this list goes on. A very big part of advanced mathematics relies on the simple rules that set theory explains. You could regard it almost as important as calculus! (although some might not agree Please Register or Log in to view the hidden image! ) I hope this explains some stuff.
All maths books are the same. You start off by going "How is this relavent to anything", or "you don't need to be a genius to work thi out", but it serves you well to meander through the first few chapters to get to the good stuff. A lot of books I have been reading start off with a kind of warning in the preface saying that the author knows the first few chapters are as boring as hell, but get through it as the rest of the book is going to rip your balls off, hehehehe
Some one beat me to it. I was going to post on Set theory pretty soon, but i wasn't ready yet, oh well. I just got a book on abstract math. I just got through all the basic defintions. I have skimmed the rest of the chapter on set theory, and it looks like it gets a lot harder. Chapter after that is on group theory. Very exciting.
I got a book about it and I've started to learn it. I guess I can see how this stuff could get a lot harder. Also, I want to learn about topology and my undertanding is that a basic knowledge of set theory is necessary to understand topology. I'm not even sure exactly what topology is, but it sounds a little like geometry, but it's not concerned with distances or measurements. It sounds interesting.
I want to learn topology too. It is soooo cool. Sometimes I look around at www.mathworld.wolfram.com I found some cool topology stuff there with 3d models and stuff. But most of it is over my head. It is still fun to look though. I would like to add at random here that I think mathematics is a form of art--it is poetry to me. It is truly a beautiful thing, and it saddens me that so few people see this.
