The difference between discrete mathematics and other disciplines is the basic foundation on proof as its modus operandi for determining truth, whereas science for example, relies on carefully analysed experience. According to J. Barwise and J. Etchemendy, (2000), a proof is any reasoned argument accepted as such by other mathematicians.
The calculus sequence dealt with real-valued functions quite well. But it did not deal with much math other than the mathematics of real numbers. Since the real numbers are continuous, this mostly left areas of math that dealt with discrete as opposed to continuous sets of values.
For example, logic deals with two values, true and false. Number theory deals with natural numbers or integers. Here are some topics covered in Discrete Mathematics, and some examples as to their importance:
1) Logic. Used in proofs to show that one step follows from the previous step. Logic is used in programming langusages, and there is even a programming language whose main purpose is to deal with logic: Prolog. Surprisingly, logic gates are used in the hardware of all modern computers.
2) Set theory. This is an unexpectedly complicated theory, given that we mostly think of a set as an unsorted collection of unique objects. What’s so hard about that? You wouldn’t believe it if I told you. One fun and important topic in set theory is that some infinite sets are bigger than others.
3) Combinatorics (basically counting). Do you know how to count? If so, if I have a class of 50 students, in how many ways can I choose 5 of them to help me move a table? If that’s easy for you, try this. McDonalds has 50 items on its menu. How many food orders with 8 items are there? (Keep in mind that repetition is allowed here. For instance, you can get 5 McRibs, and 3 large fries. A perfect meal.)
4) Recurrence relations and recursion. Recursion is a fascinating topic, and its importance in mathematics cannot be overstated. Here’s a simple question that is more easily expressed using a recurrence relation. The half-life of Cesium-137 is 30 years. If you start with 1 Kg. of Cesium-137, how much will be left after 1000 years?
5) Graph Theory. So many aspects of real life can be modeled with graphs that graph theory experts are usually focused on just a few areas. Here’s one important question that can be answered by graph theory. Given 20 cities, and the distance between them, what is the shortest route from one of the cities to another of them? A simple-to-understand problem. Not so simple to solve.
6) Tree theory. Did you know that the files and folders on your C: drive can be modeled with a tree? Or that a binary search tree can find a value in a sorted structure with 1 trillion elements before you can blink your eyes?
Goals of Discrete Mathematics:
- Mathematical Reasoning
- Combinatorial Analysis
- Discrete Structures
- Algorithm Thinking
- Applications and Modeling