This is my discrete mathematics blog. Here I will go through the topics and concepts of discrete mathematics, and I welcome you to come along, learn with me, and ask questions along the way!
Why are you doing this?
My university requires a class called Discrete Mathematics for my major. I've taken it two times now, and both times I've done poorly. Not only is this a blow to my GPA, it's taken a major hit out of my self-esteem. In order to make up for both of these things, I have decided to systematically go through the textbook (and other resources) and learn the material on a deep, comprehensive level. My goal is to go through the material that has been covered in the lectures and gain enough understanding that I could pass the class when I next take it, and also help others going through trouble with it.
What is discrete mathematics?
Discrete mathematics is a branch of math which seems to be gaining popularity. I couldn't find a good definition for it, but you might say that it's about counting things - counting how many objects are in a group, how many ways you can order things, etc. It's called discrete because it's not continuous. Let me explain that: I'm assuming, if you're looking for discrete math, you've probably got some experience with calculus. Calculus is continuous because you're playing with lines and graphs and things of that sort. Discrete math is discrete because you're just dealing with a finite number of objects, not entire lines. This will make more sense as time goes on.
On this blog, two main areas of discrete math will be looked at:
- Combinatorics - how many ways you can order things. Example: if you've ever done something like try to figure out how many ways you can order the letters A, B, and C, you've done a combinatorical problem.
- Graph theory - describes how things are related to each other. Example: have you ever looked at a map to see what roads to take to go from your home town to another city? The towns and streets make up a graph, and you were looking for the shortest path from your town to another.
What textbook are you using?
I will mainly be using Introductory Combinatorics, 5th Edition by Richard Brualdi. This is the book I've used for the last two semesters. I personally think it's a bit of a dry read. I will also dive into Discrete Mathematics and Its Applications, 7th Edition by Kenneth H. Rosen for a few chapters. It strikes me as more user friendly, but it doesn't have quite the same mathematical depth.
How will you do this?
I'm a concrete thinker. While I can handle things that are theoretical or abstract, it takes me a bit longer to really absorb them. Discrete math is a field that deals with a lot of theory. This is one of the things that put me at such a disadvantage in the two times I took the class. Plus I'm also not very quick when it comes to math beyond algebra and geometry. Therefore this blog is going to be decidedly detail-oriented. When I feel like things start to get too abstract, I'll be sure to include a small example using actual numbers. By doing that you can better get a "feel" of how the theory plays out. Concrete examples help me visualize how things work, and I imagine it will help others as well.
I'm also going to work on the problems at the end of each chapter to Introductory Combinatorics. That's because math is one of those fields that are best learned by practice and repetition. Particularly when you're a slow learner for it like me. I might also scour the Internet for other problems and resources.
Using subscripts and superscripts gets messy fast. I'm going to try and use LaTeX, a program that lets you type in code that turns text into something more mathematical looking.
Prerequisite Knowledge
If you're looking into discrete math, I'm going to assume you know algebra and calculus. There won't be a lot of calculus involved, but it's good to know for some sections. It also helps to be generally well versed with how to do math. You don't need to be a genius with it, but most problems will require you do a specific thing to reach the answer, and if you don't know the right property to use, it can be a headache.
Discrete Math and Time
If there is something I've learned about working discrete math problems, it's that they are definitely not things you can do in just a couple minutes. Usually the questions are going to require multiple steps to reach your intended goal. Occasionally, and rather frustratingly, they seem to rely on knowing some obscure fact about the numbers and variables at play. I assume this improves over time as you become familiar with the material and the theories.
Topics to Cover
Here's a list of some of the topics that will be covered on this blog.
I'm also going to work on the problems at the end of each chapter to Introductory Combinatorics. That's because math is one of those fields that are best learned by practice and repetition. Particularly when you're a slow learner for it like me. I might also scour the Internet for other problems and resources.
Using subscripts and superscripts gets messy fast. I'm going to try and use LaTeX, a program that lets you type in code that turns text into something more mathematical looking.
Prerequisite Knowledge
If you're looking into discrete math, I'm going to assume you know algebra and calculus. There won't be a lot of calculus involved, but it's good to know for some sections. It also helps to be generally well versed with how to do math. You don't need to be a genius with it, but most problems will require you do a specific thing to reach the answer, and if you don't know the right property to use, it can be a headache.
Discrete Math and Time
If there is something I've learned about working discrete math problems, it's that they are definitely not things you can do in just a couple minutes. Usually the questions are going to require multiple steps to reach your intended goal. Occasionally, and rather frustratingly, they seem to rely on knowing some obscure fact about the numbers and variables at play. I assume this improves over time as you become familiar with the material and the theories.
Topics to Cover
Here's a list of some of the topics that will be covered on this blog.
- Basic set theory
- Permutations
- Combinations
- Multisets
- Pigeonhole Principle
- Ramsey's Theorem
- Induction
- Binomials
- Multinomials
- Pascal's Triangle
- Inclusion-Exclusion Principle
- Dearrangements
- Important number sequences
- Generating Functions
- Recurrence Relations
- Graphs
- Eulerian paths, cycles, and related concepts
- Hamilton paths, cycles, and related concepts
- Bipartite graphs
- Graph colorings
- Networks
- And a few other things here and there...
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