Notes & Texts on Sets, Relations and Functions

[Math Amateur]

On a post involving the proof of the Fourth Isomorphism Theorem for vector spaces (in which I was immeasurably helped by Deveno) I have become aware that my knowledge of sets and functions was not all it should be when it comes to things like inverse images, left and right inverses and the like ...

However, I have had some difficulties in sourcing a good text that covers sets and functions in sufficient depth and detail ... many texts leave off at the point where my knowledge gets a bit suspect ...

Another area where I would like a clear and detailed exposition is indexed families of sets ... ...

Can anyone help with good texts or online notes covering these topics in some depth?

Peter

 

[Ackbach]

You're probably best off with Naive Set Theory, by Halmos, or Axiomatic Set Theory, by Suppes. The Halmos book is more informal, while the Suppes book is very formal. I know the Halmos book has indexed sets; I can't remember off-hand whether Suppes does or not. My copy is at work. I would guess that it probably does - Suppes is usually the standard reference whenever just about anything comes up.

 

[Evgeny.Makarov]

My guess is that at your level consulting Wikipedia is sufficient. It's not like the inverse function is a completely new concept to you, you probably just need to verify some details, such as if it is true that every injective function has an inverse (no). There is also an article about indexed families. If Wikipedia is not sufficiently strict, you can consult Wolfram MathWorld, Springer Encyclopedia of Mathematics and PlanetMath.org (you may need to search it using Google as in "inverse function site:planetmath.org").

I think that naive set theory is mostly common sense. I have never studied it formally. And axiomatic set theory has a different concern: to show what can be done starting from specific axioms. Thus, axiomatic set theory is not necessarily used in the rest of mathematics.

 

[Math Amateur]

You're probably best off with Naive Set Theory, by Halmos, or Axiomatic Set Theory, by Suppes. The Halmos book is more informal, while the Suppes book is very formal. I know the Halmos book has indexed sets; I can't remember off-hand whether Suppes does or not. My copy is at work. I would guess that it probably does - Suppes is usually the standard reference whenever just about anything comes up.

Thanks Ackbach ... Appreciate the help ... Will get a copy of Suppes, I think ...

Peter

- - - Updated - - -

My guess is that at your level consulting Wikipedia is sufficient. It's not like the inverse function is a completely new concept to you, you probably just need to verify some details, such as if it is true that every injective function has an inverse (no). There is also an article about indexed families. If Wikipedia is not sufficiently strict, you can consult Wolfram MathWorld, Springer Encyclopedia of Mathematics and PlanetMath.org (you may need to search it using Google as in "inverse function site:planetmath.org").

I think that naive set theory is mostly common sense. I have never studied it formally. And axiomatic set theory has a different concern: to show what can be done starting from specific axioms. Thus, axiomatic set theory is not necessarily used in the rest of mathematics.

Thanks Evgeny ... Appreciate your thoughts on this matter ...

Peter

 

[rezeew]

,

I am glad to see that you are recognizing the importance of having a strong understanding of sets and functions when it comes to more advanced mathematical concepts such as the Fourth Isomorphism Theorem for vector spaces. It is important to have a solid foundation in these fundamental concepts before diving into more complex ideas.

I would recommend starting with a textbook on discrete mathematics or mathematical proofs, as these often cover sets and functions in detail. Some popular options include "Discrete Mathematics and Its Applications" by Kenneth Rosen and "How to Prove It" by Daniel Velleman.

For a more focused approach, you may also want to look into textbooks specifically on set theory or mathematical logic. "Set Theory: An Introduction to Independence Proofs" by Kenneth Kunen and "A Course in Mathematical Logic" by Yu. I. Manin are both highly regarded texts in this area.

In terms of online resources, I would suggest checking out lecture notes from university courses on discrete mathematics or mathematical logic. MIT OpenCourseWare and Khan Academy both have free online courses that cover these topics. Additionally, websites such as Math Stack Exchange and MathOverflow have a wealth of information and discussion on these topics.

Finally, when it comes to understanding indexed families of sets, I would recommend looking into textbooks on abstract algebra or topology, as these fields often deal with indexed families in their proofs and examples. "Abstract Algebra" by Dummit and Foote and "Topology" by James Munkres are both excellent resources for this area.

I hope these suggestions will help you in your quest for a deeper understanding of sets and functions. Remember to take your time and practice, as these concepts can be challenging but are essential for higher level mathematics. Good luck!