This week we looked at functions.
- How to define a function as a subset of a Cartesian product of two sets.
- How to define a function as an explicit transformation of an element of one set into an element of another set.
- How to define a function in a Venn-like diagram.
- What it means for a function to be one-to-one, or injective.
- What it means for a function to be onto, or surjective.
- What it means for a function to be both one-to-one and onto, or bijective.
- How to prove a function is injective, surjective or bijective.
- How to prove a bijection has a unique inverse.
- How to compose two functions to create a third.
- How to prove that the composition of two injective functions is injective.
- How to prove that the composition of two surjective function is surjective.
- How to prove that the composition of a function and its inverse is equivalent to the identity function on the domain of the function.
- How to define the cardinality of any set, finite or infinite (two sets have the same cardinality if and only if there exists a bijection from one to the other).
- How to show that a set is countable (find a bijection from the set to or from a subset of the the positive integers).
- How to show that a set is not countable (suppose it is and create a contradiction.)
When examining the structure of a system of mathematics we assume, define, and prove, looking for connections and abstractions. Is this the best way to learn a system of mathematics? I don’t think it’s sufficient. It may be somewhat like learning a human language by studying its alphabet, words and rules of grammar without trying to use the language in dialog. Or somewhat like learning a computer language by analyzing its keywords and expressions without writing and testing programs (a dialog with the compiler and the computer system). Not sufficient, but perhaps still necessary. Can you believe that the set of integers has the same cardinality as the set of even integers if you don’t understand the definition of cardinality which depends on the definition of bijection which describes a property of a mathematical construct called a function which can only be formally defined by an agreed definition of the word set?
Is it surprising that some students of discrete mathematics find the course too hard? Or is it surprising that some students of discrete mathematics find the course too easy?
This quarter, based on feedback from last quarter, I decided to add more interactive examples to my lectures. A number of my new examples look back on functions from continuous mathematics: linear, quadratic, exponential, logarithmic, etc. In discussing one-to-one and onto functions, I use a mix of polynomial functions that all students have seen in prerequisite courses and present general arguments, from graphs of the functions, as to why they are or are not injections or surjections. I also show, briefly, how the first derivative of a cubic function can be used to show that it is or is not an injection. This reference to a derivative engaged some of the students who had taken calculus, but, unfortunately, it panicked some of the students who had not taken calculus. I think this is another tension of teaching—how far can you step out into other, related areas of your subject to help students make connections, without creating separation anxiety?
This week I also had several students from previous classes ask me questions, in the study center, on notation that confused them. After an exchange of questions and answers—to help me understand what they did not understand—it became clear to me that the students could see the trees but not the forest. Unfortunately, it did not become clear to them that this was the case—I couldn’t get them to stop circling the trees. Perhaps, it’s like looking at one of those ambiguous pictures that can be seen, for example, as either a rabbit or a bird. I saw the rabbit; they saw the bird. Maybe we needed to explore the definitions of class, order, family, genus and species (sets after all) and prove that we were, figuratively, mapping our mental expressions of two different species using two different one-to-one and onto visual functions. Or, maybe not.