Set Theory

Set theory, to me, probably constitutes the foundation of mathematics in a sense stricter than can be claimed by any other subdiscipline. At lots of first tier schools, there are graduate-level courses on set theory; at most other schools, there aren’t. I, personally, experienced my lone formal treatment of the discipline in the from of Math 3040 at Valdosta State University back in Fall 2007, and two things about this course stand out to me.

Firstly, I struggled. Hard. I eventually managed to squeeze out an A by what my professor later told me was a miracle: Basically, I did terribly all semester and then made a 100% on the final by spending the week before memorizing literally every single thing my professor had written on the board throughout. It took going to grad school for me to realize that learning mathematics wasn’t accomplished using that technique.

Secondly, I realize that the class I took was hardly a proper class in set theory. It was more an introduction to higher mathematics, and consisted of only about 3 weeks of formal set theory before we moved on to introductory proof techniques, induction, (semi-)formal logic, etc.

So basically, I’ve never had a course in set theory.

One result of this is my continued inability to know the quote-unquote fundamental set identities. I can usually figure them out with 70-ish percent accuracy, but generally speaking I struggle. If f:X\to Y and A,B\subset X, then does f(A\cap B)=f(A)\cap f(B)? f(A\cap B)\subset f(A)\cap f(B)? f(A\cap B)\supset f(A)\cap f(B)? And what about f^{-1}(A\cap B)? What about f(A\cup B)? The variations here are endless and, for some reason, I can never keep those things straight. This is a post to address that.

Before proceeding, note that this post came about because of my randomly pulling Sieradski‘s An Introduction to Topology and Homotopy off of my dusty bookshelf for the first time since – well, since ever. The introduction has a good balance of set theory, advanced calculus of the real line, cardinal and ordinal properties, etc. I think I’m going to give this book a once over during the next few days.

In any event, here are some things that I should work on remembering and so maybe other people out there will care to also. I’m assuming that the knowledge of the basic set operations (union, intersection, difference, product, etc.) are known.

Product Properties Let A,C\subseteq X and B,D\subseteq Y. Then in X\times Y, the following relations hold:

  1. A\times(B\cap D)=(A\times B)\cap(A\times D)
  2. A\times(B\cup D)=(A\times B)\cup(A\times D)
  3. A\times(Y-B)=(A\times Y)-(A\times B)
  4. (A\times B)\cap(C\times D)=(A\cap C)\times (B\cap D)
  5. (A\times B)\cup(C\times D)\subseteq(A\cup C)\times (B\cup D)
  6. (X\times Y)-(A\times B) = (X\times(Y-B))\cup((X-A)\times Y).

Summary: Products work “intuitively” with most set operations in most cases, although intersection obviously more so than union. Also, item (6) there will probably never stick with me fully.

Image and Pre-image Properties Let f:X\to Y be any function. Then for all subsets A, A_\alpha, C\subseteq X and B,B_\beta,D\subseteq Y,:

  1. A\subseteq f^{-1}(f(A))
  2. f(f^{-1}(B))\subseteq B
  3. f(\cup_\alpha A_\alpha)=\cup_\alpha f(A_\alpha)
  4. f^{-1}(\cup_\beta B_\beta)=\cup_\beta f^{-1}(B_\beta)
  5. f(\cap_\alpha A_\alpha)\subseteq\cap_\alpha f(A_\alpha)
  6. f^{-1}(\cap_\beta B_\beta)=\cap_\beta f^{-1}(B_\beta)
  7. f(C-A)\supseteq f(C)-f(A)
  8. f^{-1}(D-B)=f^{-1}(D)-f^{-1}(B).

Summary: Unions are more cooperative than are intersections or differences, and inverse images are more intuitive than (forward) images.

So there you have it. Hopefully by typing this out, I can keep it as a piece of data that’s fresh in my mind.

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