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Let $S$ be the set of the 16 sequences of length 4 in which each element is a 0 or a 1. Let $R$ be the relation on $S$ defined by $(s1, s2) ∈ R$ if and only if the entries of $s1$ can be rearranged to obtain $s2.$

Okay, so just "take a block of any 4 $0$s or $1$s, if they have the same amount of $1$s they can be paired."

For each element $x ∈ S$, define the set $[x]$ = {$s ∈ S : (s, x) ∈ R$}. Every element the set {$[x] : x ∈ S$} is a set. Explicitly list these sets.

This is where I'm confused about exactly what I'm supposed to be providing. If I try an provide an English translation, I come up with "list every possible pair where $s$ is a 4 digit block of $0$s and $1$s, $x$ is a 4 digit block of $0$s and $1$s, and $x$ and $s$ contain the same amount of $1$s." Do I have this right? It seems like a bizarrely easy and long task if that's the case, so I think I'm probably misinterpreting something in this notation.

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    No, you have that right – 2017-02-24
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    Example : $0101$ and $1100$ are two element (call them $s_1$ and $s_2$) of $S$ such that $(s_1, s_2) \in R$ : "they can be paired because they have the same amount of $1$s", – 2017-02-24
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    Yes, you have it right. Easy and tedious -- I sympathize. One picky point though -- the equivalence classes are sets, so enclose those lists in set braces. If you know a programming language, write a program to do it. Just as much work, but more fun. – 2017-02-24
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    Thus, consider a certain string $0001$ of $S$: call it $x$. List all strings that can be "paired" with it. All these strings are the elemnts of the set $[x]$. – 2017-02-24
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    It's just a very long and easy question – 2017-02-24
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    Huh...what a weird assignment. Alright, thank you for the confirmation, everyone. If someone wants to make it an answer I'll choose in a bit. – 2017-02-24
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    Actually, you don't quite have it right ... An equivalence class is not a set of pairs. An equivalence class is a maximal set of equivalent strings. There are 5 equivalence classes, $E_0,E_1,E_2,E_2,E_4$, containing strings with 0,1,2,3,4 ones, respectively. – 2017-02-24
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    I'm sorry, I haven't come across the terms "equivalence class" or "maximal set of equivalent strings" in the lectures or textbooks. I'm designating a more compact notation in the form: [$x_1$]-[$s_1$]-[$s_2$]-[$s...$]/ [$x_2$]-[$s_1$]-[$s_2$]-[$s...$] in order to save space, are you saying that's an inappropriate way to represent the problem? – 2017-02-24
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    Well, a set should have curly braces. – 2017-02-24
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    Yes, in my answer I'm explicitly listing the standard form of set notation and then the notation I'll be using. – 2017-02-24
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    Ok, fine -- perhaps post your answer, based on the hints, as an answer to your own question. – 2017-02-24

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The answer isn't very long. You are being asked to list the equivalence classes for the relation (the sets $[x]$). I.e., you are being asked to write down how the equivalence relation partitions $S$. You are not being asked to list each pair of sequences in the equivalence relation, i.e., each pair $(s_1, s_2)$ such that $s_1$ and $s_2$ have the same number of elements. The answer for the analogous questions with sequences of length $3$ is:

$$ \{ \{000\},\\ \{100, 010, 001\},\\ \{110, 101, 011\},\\ \{111\} \} $$

and not:

$$\{ (000, 000), (100, 100), (100, 010), \dots, (110, 110), (110, 101), \ldots (111, 111)\}.$$

For sequences of length $4$, you just have to list the $16$ sequences grouped by the number of $1$s.