Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I have a question regarding equivalence relations from my ring theory course.

Question:

Which of the following are equivalence relations?

e) "is a subset of" (note that this is not a proper subset) for the set of sets S = {A,B,C...}.

Example: A "is a subset of" B.

Now I know that for a binary relation to be an equivalence relation the relation must be symmetric, reflexive, and transitive.

I would initially say that e) would be an equivalence relation since the following:

e) This is not a proper subset so assume that each of the sets in S is a set equal to S. This should mean that the relation would be symmetric, reflexive, and transitive.

But since the possibility of the sets in S being proper subsets of S exist then the relation must have the following restriction: A R B and B R A (where R is the relation and A,B are elements of S) iff A = B. Would this mean R is anti-symmetric?

I would then guess that unless the relation is unconditionally reflexive, transitive, and symmetric then the relation could not be an equivalence relation. Would this be a correct assessment?

Any input is appreciated. Thankyou.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Ring Theory: Equivalence Relations

**Physics Forums | Science Articles, Homework Help, Discussion**