After an insanely difficult quarter I finally have time to write about math again. I really wanted to write more about discrete math throughout the quarter in order to help my students, but time was lacking. As it turns out, graduate level math eats up a lot of time. Fortunately I escaped my difficult class with an A- after getting the lowest score on the midterm. Hard work pays off.
This is a topic I need to write about so that I can get a better fundamental understanding while I study for the qualifying exam.
Abstract Algebra is one of the core math classes that a person encounters as an undergraduate math student. It begin with definitions and rules that put the definitions together in nice ways, just like almost any other math class. Before I jump into it though I want to talk about the importance that mathematicians put into definitions and rules.
Here is the idea: Go to a dictionary and look up any random word, then try to look up words that are used in that definition. Keep doing this and eventually you find yourself repeating things. The point is that you have to start somewhere, you cannot define every single word. The same thing happens in math. You cannot define every possible concept and so some things must be assumed. The most basic example of this is a set, which is a collection of elements. Well then what is a collection or an element? An element is a thing. What is a thing? It just keeps going, so instead of worrying about that we just make a good starting point that is intuitively clear.
Also, the definitions themselves are less important than how they interact with other definitions. An example that most people have seen although they may not remember is from high school geometry.
At the high school geometry level (at least when I was in the class so long ago) we were given these axioms which are basic rules that you assume to be true. Then we used the basic rules to build up either new rules or definitions of other things.
One of the axioms, and possibly the most famous one, is that parallel lines can never intersect. Parallel lines are two lines which have the same slope always. For example,
Now, it may be “obvious” that these lines can never intersect, but it turns out that this cannot be proven based solely on the definition of parallel and other basic rules, therefore we have to assume it to be true also in order to get the geometry we want. This obviousness is sort of the reason that it is an axiom. It is so obvious that it must be true, is a typical description of an axiom. Indeed assuming this and the other axioms of usual geometry and we get everything we want.
But if this is an assumption and not a truth based on other assumptions, then why do we have to do it this way? It turns out we don’t. We can just as easily assume that parallel lines do touch at some point. This simple difference in one axiom leads to a completely different type of geometry with many different consequences. For example if we are on a sphere, then two parallel lines do eventually touch at the north and south poles.
This huge example about geometry is just here to show how important assumptions are. They are the basis of every math discipline. We make up some rules about our definitions and then see what it creates. This idea is very important in learning abstract algebra because it is usually taught in this way.
To begin the discussion of Abstract Algebra, two things are needed. The definition of operation and the definition of a set. This is going to be an informal discussion.
A set as I’ve said many times, is a collection of things, called elements. For more details on sets, see some of my previous posts.
An operation is just what you would expect. It is something that takes an element, or two elements, and then creates something from those inputs. For example, addition which is denoted by: is an operation on the set of numbers. (Real numbers, integers, natural numbers, etc.) That is, I can take two numbers say, 3 and 5, and I can do an the operation to get a new element 8. This is an example which is easy, but math seeks to be completely general so try not to think too much about a single example.
For a general operation I will use the symbol
The first big important rule that we create involves 3 elements at once. It is called associativity.
We want that for any 3 elements in some set S, call the elements a,b, and c, that the following is always true.
What does this mean? Look at the example we know already.
Remember that a,b,c are just placeholders for some general thing, while is just a placeholder for whatever operation we are using. The idea is that putting the first two together and then the third, is the same as putting the last two together and then putting the first one with the result of the other two. It is something we want, which is why it will be part of a definition of something later on.
As a small side note one can try to imagine a place where this isn’t true, because it doesn’t have to be. I am just assuming it to be true because it leads to a lot of nice things but it can be replaced by a completely different rule in order to create a totally difference subject.