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Infinity is infinitely large, right? But is it possible for one infinite set to be larger than another?

Infinity is an important property of the real number system and its subsets. Let’s consider the relative magnitude of the infinite sets of numbers we just diagrammed and explore whether infinity can come in more than one size. See Note 3 below.

One definition of infinity says that a set is infinite if it can be put into one-to-one correspondence with a proper subset of itself. (A proper subset is one that is missing at least one element of the original set.) In one-to-one correspondence, each element in the first set matches exactly one element in the second set, and vice versa (i.e., each element in the second set matches exactly one element in the first set).

For example, if we have two finite sets, {1, 2, 3} and {a, b, c}, they can be put into one-to-one correspondence in the following way: 1 is paired with a, 2 is paired with b, and 3 is paired with c (and vice versa, a is paired with 1, b is paired with 2, and c is paired with 3). Here’s another way to demonstrate this correspondence:

Now let’s set up a similar numeric relationship between the counting numbers and the even counting numbers to show that the two sets can be put into one-to-one correspondence with each other:

By placing the numbers like this, we can see that for each element n in the set of counting numbers, there is a one-to-one corresponding element 2n in the even numbers set, and vice versa. This pattern extends infinitely.

**Problem B1
a. **How does this one-to-one correspondence show that the counting numbers are an infinite set?

**Video Segment**

Watch Tom and Doug as they put two sets of numbers into one-to-one correspondence and contemplate whether such correspondence makes the sets the same size. Watch this segment after you’ve completed Problem B1.

Did you come up with a different method to put the two sets into one-to-one correspondence?

You can find this segment on the session video approximately 8 minutes and 8 seconds after the Annenberg Media logo.

** **

Thus far, we have shown that the counting numbers are infinite. Here’s a new concept: Any set that can be put into one-to-one correspondence with the counting numbers is called countably infinite. It is infinite, but it can be called countably infinite because you can put it in one-to-one correspondence with the counting numbers.

**Problem B2**

Is the set of even counting numbers countably infinite? Why or why not?

**TAKE IT FURTHER**

Problem B3

**a.** Determine whether the set of integers has a one-to-one correspondence with the counting numbers.

Think about some way that you could “count” the integers so that they could be ordered “first, second, third, ….” Which specific integer would you consider “first”?

**b.** How does the size of the set of integers compare to the size of the set of even counting numbers?

Refer to Problem B1: If two sets are both countably infinite, how do their sizes compare?

**Video Segments
**What if we extended this process to the sets of rational and real numbers? Do you think they are countably infinite?

Watch this demonstration of how Georg Cantor proved that the rational numbers are countably infinite. Next, watch the proof that the real numbers are not countably infinite.

You can find this segment on the session video approximately 10 minutes and 53 seconds after the Annenberg Media logo.

You can read more about Georg Cantor’s work to determine whether all real numbers are countable in the Suggested Readings section.

Georg Cantor also studied irrational numbers to determine whether they were countably infinite. As you’ve seen, the real numbers consist of both rational and irrational numbers:

We know that rational numbers are countably infinite and that the real numbers are not (as you’ve seen in the video segment). But what about irrational numbers — are they countably infinite or not?

If you add two countably infinite sets, you get something that is also countable. For example, you could count them by alternating numbers from the two sets and put them into one-to-one correspondence with the counting numbers.

If, however, you add something to a countable set and get an “uncountable” set as a result, the second set must be uncountable as well. Therefore, the irrational numbers are not countable — they are uncountably infinite.

**Note 3**

It is difficult for us to think about infinity because our minds are finite, and we can’t easily see or touch anything that is infinite. Here is one way to try to imagine it. First, think about the counting numbers. We know that there is no largest counting number — that you can always add 1 to a counting number and get a larger one. This means that there is an infinite number of counting numbers. In this section, we will compare other infinite sets to the set of counting numbers.

**Problem B1**

**a. **The counting numbers are infinite because we have shown a one-to-one correspondence between the set of counting numbers (1, 2, 3, …) and a proper subset. Here the proper subset is every other counting number (2, 4, 6, …); the correspondence is that each number in the first set is doubled to find the corresponding number in the second, and every element in the second set is halved to find the corresponding number in the first set.

**b. **Surprisingly, they are the same size! They must be the same size, because of the one-to-one correspondence we have just found.**c. **The even counting numbers are a proper subset, because some numbers of the original set are missing. Specifically, the missing elements are all the odd numbers (1, 3, 5, …).

**Problem B2**

Yes, the set of even counting numbers is countably infinite since it can be put into one-to-one correspondence with the counting numbers (as seen in Problem B1).

**Problem B3
**

**Counting Numbers **1 2 3 4 5 6 7 8 9

**Integers **0 1 -1 2 -2 3 -3 4 -4

This pattern continues infinitely (though “countably” infinitely!).

**b.**They are the same size, because we have established a one-to-one correspondence between them.