Discrete Mathematics Function Topic

[Simpl0S]

I am currently taking a course in discrete mathematics. The literature used is "Discrete Mathematics And Its Applications by Kenneth H. Rosen" 6th ed., or 7th ed. I have encountered most of the topics from that book. I.e. Logic, naive set theory, &c. What I have encountered also is the definitions of functions in terms of images and mappings. But I always encounter this subject like some small sub-chapter in books. I.e. in the above mentioned book the following is illustrated

To what area of mathematics does this sort of symbolism and illustration of functions belong? Is there a book which you recommend on this subject? Like i.e. in courses which I have had and in the discrete mathematics course, set theory is barely touched upon, so I ended up buying Axiomatic Set Theory by Patrick Suppes to have a broader treatment of the subject. But for the functions as images (Sorry that I do not have name for this subject) I do not know where to start searching for "deeper" treatment. I feel as if these books only teach me bits and pieces from functions as mappings and I want a serious treatment on these, if possible.

I have also encountered mappings in S. Lang's Intro to linear algebra.

Also do you recommend any books on discrete mathematics? The covered subjects in my course are:

Logic and Proofs
Sets and functions
Algorithms, Integers and Matrices
Induction and Recursion
Counting
Advanced Counting Techniques
Logistic maps
Celullar Automata
Relations
Graphs & Trees (Hamiltion and Euler circuit)

According to the syllabus, from Logics until Counting it is only a repetition and a fast treatment of the subjects and starting with advanced counting techniques until graphs and trees is the main focus of this course.

 

[Stephen Tashi]

The study of functions becomes more elaborate when we restrict ourselves to considering functions that "preserve" some structure - i.e. the situation where sets of things in the domain satisfying a certain property are always mapped to sets of things in the co-domain that satisfy a similar property. Terms associated with this general concept are "homomorphism", "isomorphism", "automorphism". More specialized examples are "linear transformation", "isometry".

For some fancier diagrams, see "commutative diagram" https://en.wikipedia.org/wiki/Commutative_diagram

 

[Simpl0S]

Thank you for the reply. So if I want a deeper treatment of this subject then I have to look in the field of category theory? Any recommended books on that subject?

 

[Stephen Tashi]

The typical recommendation of a teacher for a typical student would be for the student to begin by studying concrete examples of homomorphisms (e.g. in group theory) before starting the study of category theory. An ardent category theorist might insist that category theory can be taught in elementary math courses - but I don't know of any texts written to accomplish that.

I haven't read any books on category theory. I suggest you begin by looking on the web for explanations of category theory. If you find category theory too abstract, then begin by studying the commutative diagrams of group theory.

 

[StoneTemplePython]

> Also do you recommend any books on discrete mathematics?

I am partial to an old book I found on Amazon called Basic Techniques of Combinatorial Theory by Cohen. The book focuses on proving things multiple ways and extensively uses pictures and bijections. The author even makes a point of pushing inductive proofs to the appendix stating that they are generally not constructive (especially when terse) -- which unfortunately is probably right.

I wouldn't say the book is focused on set theory, but there is a very good discussion on Inclusion-Exclusion, which is one of those simple, powerful concepts that's very easy to confuse yourself with. The chapter called "Advanced Counting Numbers" *and the one called "Generating Functions") likely overlaps with your advanced counting techniques section. The final chapter is titled "Graphs" and it is quite extensive.

That said, your course looks to be a bit more computer science oriented. You might want to check out the excellent freely available notes for MIT's "Math for CS" here: https://ocw.mit.edu/courses/electri...tics-for-computer-science-fall-2010/readings/. (Note that lecture videos, etc. are just one click away if you are so inclined.)