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In trying to understand how correspondence works, there was an example in class that my prof didn't have time to through and we instead moved on, but I wanted to see how the following interaction would have worked

Here's the example, along with showing that the two sets are uncountable

I've read that an uncountable set has both a countable and uncountable subsets, so I could see that the subsets of $S$ and $P(N)$ would lead to a possible 1-1 correspondence but I fail to see how I would determine the specific subsets to use to prove the problem. Perhaps I'm missing and or overlooking a factor?

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    There is a typo in the proof that $S$ is uncountable. It should say $t_n= |s_{n,n}-1|.$ The absolute-value sign is missing. – 2017-02-23
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    Is N supposed to be $\mathbb N$? Is $S$ supposed to be $\{0,1\}^{\infty}$ = {all sequences of 0, and 1}? If so the standard 1-1 is if $M \subset \mathbb N$ then define $s_M = \{b_i| b_i = 1$ if $i \in M$ and$ b_i = 0$ if $i \not \in M\}$. – 2017-02-24
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    @fleablood yeah N is the set of natural numbers – 2017-02-24
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    @fleablood and you got the set of S right too – 2017-02-24
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    Then for any $E \in P(N)$ let $\phi(E) =\{a_i\}$ where $a_i = 1$ if $i \in E$ and $a_i = 0$ if $i \not \in E$. And let $\phi^{-1}(\{a_i\}) = \{ n \in \mathbb N| a_n = 1\}$. That's easy to show is a 1-1 correspondence. – 2017-02-24

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Your question actually doesn't seem to have anything to do with Cantor's diagonalization argument, or the fact that these happen to be uncountable sets; instead, you are trying to come up with a bijection between the infinite sequences on $\{0,1\}$ and the subsets of $\mathbb{N}$.

As a hint on how to do that: note that $A\subseteq\mathbb{N}$ can be uniquely described by, for each natural number $n$, either declaring that $n\in A$ or that $n\notin A$.

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    Is that the notation for the infinite sequences of 0 and 1? I've seen [0, 1], [0, 1), etc. Just wanting to know for the future – 2017-02-24
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    Do you mean $\{0,1\}$? No, that's the notation for the set containing the two elements 0 and 1. – 2017-02-24