So this problem already has a solution: Problem with Equivalence Relations
I'm good for the majority of it except for part c), I wasn't able to figure that out on my own or by looking at the explanation.
Perhaps I still don't fully comprehend what equivalence classes are, but from my understanding they are subsets of out general set A, but they are special because they contain a group of elements that share the same equivalence relation characteristics? I mean the question asks to describe the set of equivalence relations that contain my set S. So it is beyond the elements that would be from set S that confuse me. Why would any other sorts of elements be a part of the intersection of that group of equivalence relations if the one set they share is the set S?
Before we get into this, recall the definition of equivalence relation and equivalence class. An equivalence relation is just a way to compare two elements. If two elements are equivalent, they belong to the same equivalent class.
For c) What we are searching is a equivalence relation $\sim$ such that for all elements $\overset{\rightarrow}{x}\in S,\ \overset{\rightarrow}{x}=(x_1,x_2)$ we have $x_1\sim x_2$. This is meant by equivalence relations on the real line that 'contain' $S$. We need more elements, than the one in $S$ since the relation defined by $S$ i.e. $x\sim_s y \Leftrightarrow y=x+1, x\in ]0,2[ $ is NOT an equivalence relation.
Furthermore, we want $\sim$ to be the smallest equivalence relation containing $S$. With this in mind, you should be able to follow the ideas from your linked post and solve C.
Does this bring some clarity to what is asked?