You're a caveman, and you have two big piles of stones. And you want to know if they have the same number of stones. Now, if you wanted to do this as yourself, you'd just count one pile then count the other. But remember - you're a caveman! You don't know HOW to count!
Luckily this is solvable.
Take one rock from each pile. Throw them away. Repeat until you run out of rocks. If one of the piles runs out first, it's smaller. If they both run out at the same time, they're the same size.
So let's start with two piles. One of them contains the numbers 1, 4, 9, and 16. The other contains the numbers 1, 2, 3, and 4. We match two numbers - 1 and 1 - and throw them away. We match two more (4 and 2) and throw them away. 9 and 3. 16 and 4. Hey! We're out of numbers! AT THE SAME TIME. There must have been the same amount in each group!
So now let's take something a little more complicated. Say, "the first 2000 integers starting at 0" and "the first 2000 even integers starting at 0". And let's do the same thing.
First, we take 0 and 0. Then we take 1 and 2. Then we take 2 and 4. Then we - okay, I'm not writing this out.
We take n, right? And we match it with n*2. Obviously for every n, there's an n*2 to match it with. Precisely one n*2, you can't get two different values out of that. And for every n*2 there's precisely one n. Therefore they're the same size.
Why limit it to the first 2000 integers, though? There's no reason we can't just say "all the positive integers" and "all the positive even integers". For every positive integer n, we match it to the positive even integer n*2. And vice-versa. And clearly we cover all of both sets, with no missing values or repeated values.
Therefore, they're the same size.
Yes, that's right. There are the same number of positive even integers as there are positive integers in general. Despite the fact that one is a subset of the other.
Stop looking at me like that.
So here's a few other fun things to compare, and I'll post the answers eventually.
(1) Which is larger - the set of positive integers, or the set of all integers?
(2) Which is larger - the set of all integers, or the set of prime integers?
(3) Which is larger - the set of all integers, or the set of all integer-pairs? (i.e. (4,5), (150,-4), etc.)
(4) Which is larger - the set of all integers, or the set of all rational numbers? (I.e. numbers that can be expressed as fractions.)
(5) Which is larger - the set of all integers, or the set of all finite sets of integers?
(6) Which is larger - the set of all integers, or the set of all real numbers?
If you already know the answer, duh, don't respond - leave it up for the hundreds of people who haven't taken this in college and yet find it interesting to work on. Since, you know, I'm sure you've been dying to be able to compare infinities. It comes in so handy in so many places!