Discrete Mathematics is a precursor to Modern Algebra (also called Abstract Algebra), much like Matrix Algebra is a precursor to Linear Algebra. Some topics are briefly introduced below, and please note that this list is not comprehensive. Also, the ideas introduced below are simple, and Discrete Mathematics includes some beasts that do not live near simple. Finally, if you notice that we cover something in four pages and the textbook on the topic is over 400 pages, you can guess we glossed over a lot and skipped a lot.
- Number Theory
- Set Theory
- Proposition Logic
- Combinatorics
Number Theory
Countability
Countability is the idea that you are working with a system that you can count up to any number, at least in principle. If I give you a number, like 1,000,000 and you give me your counting speed, we can calculate how long it will take for you to get there. The point is, you can get there.
If we are working with continuous numbers, when you say “1, 2” someone yells out “Wait, you missed 1.5!” You try again and another person yells out “Wait, you missed 1.2”. Very quickly you realize that for any second number you pick for counting, there will be an infinite number of values between your starting number and that arbitrary number that you chose. You can’t count from 1 to 10. You can’t even count from 1 to 2.
A set of numbers is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. Imagine, for example, if you chose for your numbers to be {1/4, 1/2, 3/4, 1, 5/4, 3/2, etc.} We might argue for this system in a world where we had dollar bills and quarters.
Set Theory
Set Theory deals with putting like things together and doing manipulations on the sets. The idea of sets appears when we build a Hierarchy. We might discuss a set including all Animals, and then we mention that a set of all Cats is a subset of the set of Animals. We may be interested in the intersection of two sets, or alternately, we might ask, what is in the set that includes all the elements of a first set and a second set? There is a notation () used for this work, and
You will probably notice some similarity between Set Theory and the next topic, Propositional Logic. Please smile and give yourself some credit when this happens.
Propositional Logic
A study of Propositional Logic will have you discussing logic informally, using full sentences and maybe talking about an idea using several sentences to explain it, and it will also show you a formal notation that uses a variety of symbols (∧, ∨, →, ↔, ¬) and alphabet letters to represent ideas. It is a tradition to use ‘p’ and ‘q’ for the letters when discussing the basic ideas. Depending on the author of the text, the idea of an “exclusive or” might be introduced, where “a or b” requires that one of the variables is true and the other is false”.
The idea is to establish a structure representing the logic of what is being discussed. This might allow you to take one established truth and get to another truth. It might take you from an established truth to a contradiction, or said another way, you can see that in order to keep the first idea, you have to discard the second. You might be able to prove that are argument is based on circular logic, and thus it doesn’t logically prove what someone was hoping it would prove.
Work with Logic will give you the idea of how any study needs things like premises, axioms and postulates. Along with this, you should develop a respect for the need to keep these as simple and as few in number as is possible. The story of how people went back and redid math after discovering Russell’s Paradox helps to give some context to all of this.
Combinatorics
Combinatorics is the study of possible configurations or configurations. You might ask, how many possible results are there if we flip a coin twice (H for heads and T for tails). We see two ways of doing this:
- H,H
- H,T
- T,H
- T,T
or we might say that H,T and T,H are the same and make the list as
- H,H
- H,T
- T,T
It turns out both lists are possibles. In Statistical Mechanics, the list with four items is a list of microstates, and we say those come from the three possible macrostates, shown in the second list. The macrostate (H,T) contains two microstates and the other two macrostates have one microstate in each. This counting of macrostates and microstates is useful for understanding Entropy.
Latin Squares
The topic of Latin Squares falls under Combinatorics. For a Latin Square, the number of different answers equals the width of the square. Inside the square, these answers each appear once in each row and each column column.
We show an example below with the integers, 1, 2 and 3:
Rob Sterling 