TITLE: Naive Set Theory by Halmos is confusing to a layman like me
QUESTION [23 upvotes]: I want to be able to express set notations fluently in math fields used in machine learning, so I started reading Naive Set Theory by Halmos.
But I have been facing a lot of problems like :

On pages 1-6 , I encountered if and only if and had to go to Wikipedia, to actually understand it .
Also , I had to search Math.SE to understand such sentences as :



It is equally harmless if the letter used has already been used with
  "for some" or "for all". Recall that "for some $x \ (x \in A)$" means
  the same as "for some $y \ (y \in A)$" ; it follows that a judicious
  change of notation will always avert alphabetic collisions.
" nothing contains everything "



Every page is a conundrum that requires huge amounts of mental gymnastics. I think I'm not ready to read the book, yet.
Can anyone recommend a better introduction to informal set theory than Halmos ?
My Background : I majored in biology and attended calculus, statistics, real analysis, and linear algebra classes in a university. However, I dropped out of all math classes very early to focus on biology. Now, I want to learn math again. The high school didn't teach any theoretical foundation of math at all.

REPLY [5 votes]: I think you're reading the wrong book. Of course, if you want to continue reading Halmos for fun and/or the intellectual challenge, that's great, but what you're doing is the equivalent of trying to learn to drive by reading books about automotive engineering. A more mathematical analogy would be to think back to the courses you took on calculus and real analysis: the calculus course was about how to use differentiation and integration to solve problems, whereas the analysis course is about how differentiation and integration work "under the hood".
For machine learning, you will be a user of sets and you might do some calculation on sets but you won't be doing research into what sets are and what are the limits of what they can do. For your purposes, you don't really need to know what's "under the hood" and it will be entirely sufficient to say that a set is a collection of objects defined by some stated property. Yes, that way lies Russell's paradox but Russell's paradox only comes up if you use "weird" properties to define sets and you won't be going anywhere like that. In particular, essentially all the objects you'll come across that you might want to describe as "sets" will be subsets of something that definitely is a set, such as the integers. In those areas, set theory works in just the way you'd expect it to.
The first chapter of any undergrad textbook on general discrete mathematics will almost certainly contain all the set theory you need for machine learning.