Lecture 1: Comparing Things in Sets

We begin by talking about a set. Simply put, a set is a collection of things, any sort of thing at all, we could look at the set of stars in the sky, the set of purple cars, or the set of yellow polka dot bikinis and the people who wear them. Mostly, we will talk about sets of things like numbers or letters. For example, the set that consists of A, B and C is perfectly good set. So is the set that consists of 1, 2 and 3. To make things easier, we will often write the list of things in a set with curly brackets, so we would write {1, 2, 3} for the most recent example. The things in a set are called elements.

You might have noticed that the sets {A,B,C} and {1,2,3} have something in common, they both have three elements in them. Given a set we can count the number of elements in that set. This number is the cardinality of the set. For example, the set {2, 3, 5, 7, 11, 13} has cardinality 6. So does the set {2, 4, 6, 8, 10, 12}, the set {1, 1/2, 1/3, 1/4, 1/5} and (not so surprisingly) the set {1, 2, 3, 4, 5, 6}.

Seeing if a set has 6 elements in it seems pretty easy, but what if you needed to see if a set had cardinality 100? What about a million? You might not want to count that high if you were trying to count the elements in a set. Lets return to our earlier example to try to figure out an easier way to see if two sets have the same cardinality. We can try to find out if two sets have the same cardinality using something that looks like the buddy system. If we can match two sets up so that each element in the first set has exactly one "buddy" in the second set, and every element in the second set is the buddy for some element in the first set, then they must have the same number of elements. We will also make sure that when we start with one element in the first set, find its buddy int he second set and then find the element in the second set's buddy in the first that we end up with the original element with which we started. Let's take the sets {2, 3, 5, 7, 11, 13} and {1, 2, 3, 4, 5, 6}. They both have cardinality 6, so if out buiddy system plan works we must make sure that the sets can pair off. There are a lot of choices about how to do this, for example:

 
2  3  5  7 11 13
|  |  |  |  |  |
1  2  3  4  5  6
Or alternately:
2  3  5  7 11 13
|  |  |  |  |  |
6  5  4  3  2  1
Or any other matching that you dream up.

If you can find this sort of matching for really big sets, then you can be assured that they have the same cardinality, or number of elements. These matchings, where each element in a set has exactly one buddy element in another set is called a bijection.

Exercises

  1. What is the cardinality of the set of cars parked on the block in front of the building you are currently in?
  2. What is the cardinality of the set of kids in your class? Is it possible to find a bijection between kids in your grade at your school and kids in the next grade? Why or why not?
  3. Find a bijection between the set {A,B,C,D} and the set {Z,Y,X,W}.



This work was made possible through a grant from the National Science Foundation.