Reddit reviews A Book of Set Theory (Dover Books on Mathematics)

I don't know where to even start :)

Infinity is a property. Sort of like an adjective. You don't say something is an infinity, but rather something is infinite in size. Think of "infinite" rather than "infinity".

A set is a collection of unordered objects(anything at all), like so

{shoe, car, &, 3}. It's unordered because we can write it as {car, 3, shoe, &}.

A set is said to be finite if we can pair everything in it with another reference set {1, 2, 3, ..., n}.

In the reference set "..." means that numbers continue in the given order. "n" at the end means that there's a number at which the numbers terminate.

Lets call our set {shoe, car, &, 3} A. So, A = {shoe, car, &, 3}.

Now compare the elements(objects) inside A with those inside {1, 2, 3, ..., n}:

shoe can be pared with 1.

car can be pared with 2.

& can be pared with 3.

3 can be pared with 4.

So everything in A is pareable with everything in {1, 2, 3, ..., n}.

So, A is finite according to our definition above.

Definition: S is an infinite set if and only if there exists a set A such that A is a proper subset of S and |A| = |S|.

Ok. This is one of the definitions of infinite set and to understand that you need to be familiar with the notions of functions, mapping, cardinality, bijection, equality, existence, proper subset...all pretty basic notions.

Tell you what, why don't you just study these books below that would teach you all about these notions and much, MUCH more?

A Book of Set Theory by Charles Pinter.

Naive Set Theory by Paul Halmos.

The books above will not only teach you about finite/infinite sets, but also can serve as a very nice foundation to study higher math.

If you'd like to read up on set theory, there are plenty of reasonably priced books available, particularly from Dover. Here is one such book. I don't own this book in particular, but Dover books are usually pretty decent given their extremely low price point, and aren't usually written to be dense as bedrock. That being said, don't expect to absorb everything through osmosis by just skimming the text. If you really want to learn and absorb the material, you will have to sit down, read, re-read, and work, but it's only an insurmountable task if you tell yourself it is. If you have any questions, or need help or insight, you can feel free to ask them here on /r/math (though they'll probably be best put in the simple questions sticky) or over at /r/learnmath.

The same Pinter who wrote the much lauded and dirt cheap Dover text on introductory abstract algebra recently came out with a book on set theory, also a dirt cheap Dover text.

It looks really great and covers quite a bit.

Yeah, I love the intro to that book. The intro to his other text, A book of set theory, is a great history of the foundations of mathematics as well. I would definitely recommend either of Pinter's books for this purpose, they are self contained and nicely motivated.

I haven't used the set theory books myself so I can't comment on their quality, but anytime I hear someone looking for reasonably priced math books I immediately think of the Dover Books on Mathematics series.

https://www.amazon.com/Theory-Logic-Dover-Books-Mathematics/dp/0486638294

https://www.amazon.com/Axiomatic-Theory-Dover-Books-Mathematics/dp/0486616304/ref=pd_bxgy_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486616304&pd_rd_r=c84f7c07-350b-4ec3-8fc5-adf30ae9b20c&pd_rd_w=74PSR&pd_rd_wg=RG95z&pf_rd_p=a2006322-0bc0-4db9-a08e-d168c18ce6f0&pf_rd_r=RKXGP4020J1B5GVED0PH&psc=1&refRID=RKXGP4020J1B5GVED0PH

https://www.amazon.com/Book-Theory-Dover-Books-Mathematics/dp/0486497089/ref=pd_sbs_14_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486497089&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96

https://www.amazon.com/Naive-Theory-Dover-Books-Mathematics/dp/0486814874/ref=pd_sbs_14_1/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486814874&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96

edit: added more books

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis