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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?

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D.C. the III
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1 Answers1

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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?

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