0:00
Hi.
00:02
In this video, we want to find a bijection that proves that the set of integers is countable.
00:07
So what this means is we want to find a function f from the natural numbers n, which it depends how some people define them differently.
00:19
We'll define it as a 0, 1, 2, and so on.
00:28
Sometimes people don't include 0 to the integers z, right, which is.
00:36
Negative 2, negative 1, 0, 1, 2, so on.
00:47
So we want to find a bijection, which means it's injective and surjective.
00:54
So let's just list the natural numbers.
00:58
We have 0, 1, 2, 4, and so on.
01:08
So we want to associate with each one of these natural numbers, a unique integer.
01:14
So what we'll do is we'll first assign 0 to 0, and then, well, i guess we can just assign, we'll let's assign 1 to negative 1.
01:32
Okay, so for each even number, we want to assign all of the positive integers.
01:41
So what we can do to get all even numbers onto the into all positive integers is we just divide it by 2.
01:49
So two goes to one, four goes to two, six goes to three, and so on.
01:57
And then three, we're basically going to do the same.
01:59
I mean, sorry, with odd numbers, we're basically going to do the same thing, except that we have to get the negatives now.
02:06
So what we're going to do is we're going to add one to the number, divided by two, and then make it negative.
02:18
So in the case of three, we add one to get four divided by two.
02:23
And then make it negative.
02:25
So negative 2, 5 goes to negative 3 and so on.
02:30
So this looks good, right? you get 0, negative 1, negative 2, negative 3, 3.
02:36
So it's clear that this is surjective, and it's also clear that it's injective.
02:42
But let's write out what the actual function definition is just for concreteness.
02:49
So i'll use n instead of x because it's just natural numbers...