- #1
wubie
Hello,
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.
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.