 Alright, let's take a look at problem 111. So this is to prove the cardinality of the set of whole numbers is infinite. And again, this is a proof problem, which means we have to go back to our definition. In this case, we have to go back to our definition of what it means for the cardinality of a set to be infinite. And you should review the definition of infinite cardinality. And again, problem solving means creating your own solution. Try to solve this problem before watching the rest of the video. Because if you watch the rest of the video, at that point you will never have an opportunity to solve this problem or really any problem that requires you to prove the cardinality is infinite. The cardinality of any set being infinite is going to be an extension of what we do in this video. So if you watch the rest of the video without doing that, you will never again have the chance of solving this particular type of problem. So if you want to proceed, then remember that infinite cardinality requires finding a one to one correspondence between our set and some proper subset. So our set of whole numbers and a proper subset of the whole numbers. So to start off with, let's go ahead and just put down our set of whole numbers in list notation, partial list notation anyway. So it's the numbers 1, 2, 3, 4, 5, and so on. We write that, we indicate that using our dot, dot, dot ellipses. And what we want to do is we want to find a proper subset. And so a good way to start here is to think about how we might classify the whole numbers. So I have these numbers here and I might think about, well, how I would classify these numbers. And so this is a number and it has some property and this is a number and it has some property and so on. And so we might go through our list of properties of whole numbers. And one possibility is to consider stopping the video and trying to come up with your own possibilities. Because again, as soon as you see how this is done, you will never have the chance of solving this problem and you will never get this opportunity again. Again, problem solving opportunities in real mathematics courses are extremely rare. If you're lucky, you might have 20 problem-solving opportunities in the course of a semester. Well, so after some thought, we might decide, well, one classification of whole numbers is even numbers. Whole numbers are either even or not even, even or odd. And so maybe I'll do the same thing. I'll list those even numbers. There's 2, 4, 6, 8, 10, and so on and a whole bunch of others. And so now let's think about this. Here's a set of even numbers. Our set of whole numbers. And we do want to check to see if we actually have a proper subset. Is the set of even numbers a subset of the set of whole numbers? Is everything in here someplace up here? Well, 2 is here, 4 is here, 6, wait, 6 isn't up here. Oh, wait, remember the dot, dot, dot says there's more stuff here that we're not writing. 6 is a whole number. So it's going to be up here someplace. 8 is a whole number. It'll be up here someplace and so on. So everything in our set of even numbers is someplace in our set of whole numbers. And we want to verify whether it's a proper subset. These two have to be different. Well, here's our set of whole numbers. Here's our set of even numbers. These aren't the same thing. This set includes things that aren't down here. One isn't down here. Three isn't down here. And so on. So this is a proper subset. Now for the challenge, we want to try and find a 1 to 1 correspondence between the two sets. I want to find some way of matching every whole number with an even number and vice versa. So, well, the obvious thing to try and do is match 1 to 2 and so on. So I might try and match this way. And let's see. Well, we do have to check, does every whole number get matched to a unique even number? So I have this whole number 3 gets matched to 6. If I have another whole number over here, 11, for example, if I have the number 11, what's it going to get matched to down here? Does it get matched to something? And if you think about it, yeah, 11 will get matched to an even number, 22. And conversely, is every even number going to be matched to a unique whole number? So if I have an even number, 26, does it get matched to a unique whole number? And again, if you think about it, that 26 should get matched back up to the number 13. So there's our 1 to 1 correspondence. We do need to summarize it. So we'll point out a couple of things. First of all, the even numbers are a proper subset of the whole numbers. That's what our infinite cardinality requires. And every whole number can be matched to a unique even number, namely, twice whatever the whole number is. So I'm going to take the whole number 3, and I'm going to match it to 6. I'm going to take the whole number 5, match it to 10. I'm going to take the whole number 157, and match it to whatever twice that number is. And so here's our rule that defines our correspondence. And also every even number is going to be matched to a unique whole number, half of what the even number is. So 10 gets matched to half of 10, that's 5. The even number 47, well, that's not even, the even number 46 gets matched to half of what that number is. So 46 gets matched to 23. So every even number is matched to a unique whole number. And so now I have my 1 to 1 correspondence between the set of whole numbers and a proper subset. And that's what my definition of infinite cardinality requires. And so my set of whole numbers has infinite cardinality. Now the proof is all of this. You have to write all of this down to have a complete proof. If you omit any portion of it, your proof is not complete. You want to show that the correspondence exists, and really this is the important part. This is just an illustration. The important part of the proof is actually these two paragraphs.