Lecture 2: Bijections

In the last lecture we talked about bijections to figure out whether two sets had the same cardinality. Now lets try to do that with a pair of sets where we know that we cannot actually count up all of the elements of the set, no matter how long we spend counting. We will look at infinite sets, sets with more elements than any number you can count up to, no matter how long you spend counting. One such set is the natural numbers, that is, the numbers 0, 1, 2, 3,... (we will use the ... to mean "keep going with this pattern"). Another example is the positive numbers, the numbers 1, 2, 3,... . At first glance, you might say that the positive numbers should have cardinality one fewer than the natural numbers, since they are just the natural numbers without zero.

But here is a problem: what is the infinite cardinality of the natural numbers that we can't count up to minus one? Well, maybe bijections will help us sort out this sticky situation. In the last section, we talked about how if there is a bijection between two sets, then they must have the same cardinality. In this case, we will find a bijection between the naturals and the positives and then conclude that they must have the same cardinality. The bijection that I propose matches each natural with itself plus one in the positives, that is, we match n in the naturals with n+1 in the positives. In the pictures from the last section, this looks like:


0  1  2  3  4  5  ...

|  |  |  |  |  |  |||

1  2  3  4  5  6  ...

Now, it might seem worrisome that we could run out of room to keep shoving the natural numbers over one to make room to find 0 a buddy in the positive numbers, but that is the joy of an infinite set, we always have more room, just ask for another number!

Now let's try for a more complicated bijection between infinite sets, by looking at two sets that appear more different, the naturals and the even natural numbers. At first thought, you might think that there should be half as many even numbers as only every other one of the natural numbers is even. As we try to investigate potential bijections, let's start by trying to understand the even numbers a little bit better, by asking ourselves why it means to be an even number. The standard definition of an even natural number is a natural number that we can divide by two and still get a whole number, with out fractions, that is, another natural number. Now, lets say a number, call it x, can be divided by two to get another whole number. Then, x/2 = some whole number. Let's call this number n. Then, re-arranging our first equation, we can re-write our even number, x, as 2n, i.e. x = 2n. So now, since x was just any old even number, we can write every even number as exactly 2n, where n is some natural number. This suggests some sort of natural bijection, since we see that every even number is exactly twice a natural number. From a natural number, we can find an even number by multiplying by 2, and we can get back to the original natural by dividing the even number by two. So the bijection we have looks like:


0  1  2  3  4  ...

|  |  |  |  |  |||

0  2  4  6  8  ...

Exercises

  1. Can you think of any other bijections, that is different matchings, between the natural numbers and the positive numbers?
  2. Can you find a bijection between the odd numbers and the natural numbers? (Hint: try to write out what each odd number looks like the same as we did for the evens.)



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