When I visit family for the holidays, the topic of my being a mathematician always seems to come up, and there’s always a child in the family struggling with maths, and when I ask the subject of their struggles the word “algebra” is always the culprit. I’ll save for another post my ideas on how this subject should be taught in high school and some of the main problems facing students. I want to concentrate this post on a topic that few outside the mathematical world know about, but which many inside this world (myself included) hold dearly – the topic of modern or abstract algebra. I refer to this as “the other” algebra because a general conversation about the word “algebra” will generally revolve around concepts such as systems of equations, slopes, intercepts, intersection, rise-over-run, point-slope, and other terminology that limits algebra to a specific domain (the set of real or complex numbers) while at the same time ignoring the underlying beauty associated with this area. I wrote previously about the area of set theory and the beauty associated with taking math out of the scope of a basic number line and into a much more undefined space. Abstract algebra is a continuation of set theory where in addition to our set, we have a (binary) operation defined on any two elements of this set. The inclusion of this binary operation allows us to consider several different structures based on the properties that this binary operation holds. The structures I’d like to write about today are called groups. A group is a set along with an operation (or function) defined on any two elements of the set with the following properties: – It is closed. This means that any time we run this function on two elements on the set, the function gives us a member of the set. In mathematical terms, for all a, b in the set A, f(a, b) must also be a member of A. – There is an identity element. An identity element is defined as an element where is we include it in the binary operator with any other element, the operator will always return the other element. So if the element i is the identity element, then f(i, a) = a and f(a, i) = a for any other a in the set A. Any group must have an identity element. – Every element has an inverse. Inverse elements are based on the identity element. What the inverse says is that for every element, there is a way to use the binary operator to get to the identity element. So for all elements a in the set A, there is an element b in the set A such that f(a, b) = i, where i is the identity element. – The binary operator is associative. I described the associative property when I discussed the functions and relations of set theory. A function is associative if the way we group things (aka associate them) doesn’t matter. This means that for any elements a, b, and c of the set A, f(f(a, b), c) must be the same as f(a, f(b, c)). If these four properties hold for a set A and a binary function f, then we say that the pair (A, f) is a group. We will generally use a common notation such as a · b, or a * b or simply ab to represent f(a, b). Another important concept in group theory is the idea of a Cayley table. These are similar to multiplication tables that we drew out when we were first learning our “times tables”. For a group with n elements, we form a table with n rows and n columns. Each element of the group is written out to the left of each row and above each column (so really we can think of it as an n+1 by n+1 table with the first row and column being descriptive rows). Each cell of the table is the binary operator applied to the two elements indicated by the row and column (with an understanding of whether we have row before column or vice versa). Obviously, we can only do this for finite groups as we cannot write out all the elements of an infinite set. The script I’ve added is a tester to allow users to input the information for a possible group (size, name of each element, and a Cayley table) and with this information the user is informed whether or not it forms a group. If it does not form a group, the reasons why it does not form one are also given. There are also some sample groups given to give insight into this area.
I have just finished a script that helps users understand mathematics through set theory and relations. Much of our world deals with relationships – both in the sense of romantic ones or ones that show some interesting property between two sets. When mathematicians think of set theory, a relation between the set A and the set B is a set of ordered pairs, where the first element of the ordered pair is from the set A and the second element of the ordered pair id from the set B. So if we say that R is a relation on the sets A and B, that would mean that R consists of elements that look like (a, b) where a is in A and b is in B. Another way of writing this is that R is a subset of A x B. For more on subsets and cross product, I refer you to my earlier script work on set operations. Relations can provide a useful means of relating an abstract concept to a real world one. I think of things like the QB rating system in the NFL as an example. We have a set of all quarterbacks in the NFL (or really all people who have thrown a pass) and we would like some means of saying that one QB is performing better than another. The set of statistics kept on a QB is a large set, so attempting to show that one QB is better by showing that every year that they played one is better in every statistical category can be (a) exhaustive, and (b) will lead to very few interesting comparisons. Most of the really good QBs have some areas that they are really good and others that they are not. The QB rating system provides a relation between the set of all QBs in the NFL and the set of real numbers. Once this relation was defined, we can say that one QB is performing better than another if his QB rating is higher. Similarly we can compare a QB to his own statistics at different points in his career to see the changes and trends. This is just one example, and there are countless others that I could have used instead. Once we understand what a relation is, we have several properties that we are interested in. Below I list four, although there are many more. Properties of Relations: A relation R is symmetric if whenever an element (a, b) belongs to R, then so does (b, a). A relation R is reflexive if for every element a in the universe of the relation, the element (a, a) belongs to R. A relation R is transitive if for every pair of elements (a, b) and (c, d) and b = c, then the element (a, d) belongs to R. A relation R is anti-symmetric if the elements (a, b) and (b, a) do not belong to the relation whenever a is not equal to b. Once we understand what a relation is, there are a few common ones that we are interested in. Below I list four, but again, I want to stress that these are some of the more common ones, but there are several others. Types of Relations: A relation R is a function (on its set of defined elements) if there do not exist elements (a, b) and (a, c) which both belong to R. A relation R is an equivalence relation if R is symmetric, reflexive and transitive. A relation R is a partial order set if R is anti-symmetric, reflexive and transitive. A relation R is a total order set if it is a partial order set and for every pair of elements a and b, either (a, b) is in R or (b, a) is in R. A partial order is just an ordering, but not everything can be compared to everything else. Think about the Olympics, and a sport like gymnastics. Consider the floor and the balance beam. One person can win gold on the floor and another person wins gold on the balance beam. That puts each of them in the “top” of the order for their particular section, but there’s no way of comparing the person who won the floor exercise to the person who won the balance beam. So we say the set is “partially ordered”. More formally, lets say that two people (person X and person Y) relate if they competed in the same event and the the first person (in this case person X) received an equal or higher medal in that event than the second person (in this case person Y). Obviously any person receives the same medal as themselves, so this relation is reflexive. And if Jamie received an equal or higher medal than Bobby and Bobby received an equal or higher medal than Chris, then Jamie must have received an equal or higher medal than Chris so this relation is transitive. To test this relation for anti-symmetry, suppose that Chuck received an equal or higher medal than Charlie and Charlie received an equal or higher medal than Chuck. This means that they must have received the same medal, but since only one medal is awarded at each color for each event (meaning one gold, one silver and one bronze…if this is not true, assume it is), this must mean that Chuck and Charlie are the same person, and this relation is thus anti-symmetric. If we have a partial ordering where we can compare everything, then we say that the set is “totally ordered”. An equivalence relation tries to mimic equality on our relation. So, staying with that example of the Olympics, an example of an equivalence relation could be to say that two athletes relate to one another if they both received the same color medal in their event (for the sake of argument lets assume that no athlete competes in more than one event). Then obviously an athlete receives the same medal as themselves, so this relation is reflexive. If two people received the same medal, then it doesn’t matter if we say Chris and Charlie or Charlie and Chris, so the relation is symmetric. And Finally if Chris received the same medal as Charlie an if Charlie received the same medal as Jesse, then all three people received the same medals, so Chris and Jesse received the same medals and this relation is transitive. Because this relation has these three properties, it is called an equivalence relation.