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Section Chapter 0

Subsection 0.1 What is Discrete Mathematics

We started the course by exploring some of the typical questions that show up in Discrete Mathematics by working through the Introductory Problems activity.

Handout 0.2 Discrete Structures Notes

Before starting, what questions do you have about the syllabus?
We will come back to the problems from last time throughout the semester. Today we want to come to some agreement about the vocabulary of discrete math by considering what mathematical objects we will study. Let’s see what structures could be useful for abstracting the problems from last time.
  • Here are the types of structures we will explore this semester: Sets, Functions, Sequences, Relations, and Graphs. (And one more: statements, but that is not like the others.)
  • Introduce the notation of sets. Relate this to the handshakes problem. What we are really interested in here is a set of handshakes. What is a handshake? It is a set of two people. In particular, the “order doesn’t matter.”
  • Suppose instead we wanted to count the number of piggy-back rides that occur if everyone gives everyone else a piggy-back ride. Now we are counting how many two-element sequences there are. This is one way of thinking about sequences: a finite sequence, which we sometimes think of as a tuple.
  • Another way of thinking about sequences is an infinite list of numbers. This can be useful in other ways: by giving us a sequence of answers to different versions of questions. For example, the sequence of numbers of hotdogs people eat.
  • A sequence is really a type of function: it’s input is a natural number (the set of natural numbers is our favorite set in this class: \(\{0,1,2,3,\ldots\}\text{;}\) note it starts with 0).
  • What can we say about functions in general?
  • A function is really a particular type of (binary) relation. You can think of a function also as a set of tuples (sequences). But there are other sorts of relations, and changing the properties of a relation can lead to interesting and useful insights.
  • A particular type of relation that we will study a lot in this class is called a graph. This is like the 5 towns and roads question, although there we are also asking wither the relation that holds of the towns (that every pair of towns is connected by a road) can be embedded on a sphere in such a way that the roads do not cross. We might also have an abstract graph and not care about such an embedding.
For next class, see how far you can get on the two logic puzzles of the Logic Puzzles activity. We will discuss them together in class next time.

Subsection Chapter 1 Preview

Let’s see what doing logic means by looking at the two examples from the Logic Puzzles activity.
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