Cantor's Diagonalization Argument- Application Proof Problem

Question

$$a_k = \begin{cases} 1 & \text{if the}~k\text{th digit of}~f(k)~\text{is}~ 0\\ 0 & \text{otherwise} \end{cases}.$$

Problem: You are a consultant for a friend designing a new video-game. Every player in the game is assigned a unique ID which is a binary string with 3 digits. Currently only the following three IDs have been assigned to players: 000, 001 and 010.

Apply Cantor’s Diagonalization argument to get an ID for a 4th player that is different from the three IDs already used.

I can't wrap my head around this problem. So, the point of Cantor's argument is that there is no matching pair of an element in the domain with an element in the codomain. His argument shows values of the codomain produced in a diagonal fashion producing unique values.

How do I solve this problem? Can someone please provide an answer with an explanation?

Answer 1

Arrange the three IDs already used in this order

A 000
B 001
C 010

Now choose the next ID xyz so that you can be sure its first place x is different from the first place in A. (There is only one way to do that!). Now choose y so that the new ID is surely not B. And then the same for z.

Extra note. This is a good way to understand Cantor's diagonal process but a terrible way to assign IDs. Can you see why you can't use this strategy to make an ID now for a fifth player? A better way is to think of these IDs as binary numbers.