Thread: Equivalence relation and total ordering problem

Let R denote a relation defined on a set A as follows: xRy iff x<=y where <= is a total ordering on A. Can R be an equivalence relation on A?

Originally Posted by jsteel2

Let R denote a relation defined on a set A as follows: xRy iff x<=y where <= is a total ordering on A. Can R be an equivalence relation on A?

What if the set only contains one element.

So the argument is that R can only be an equivalence relation if the set A has atleast two elements?

Originally Posted by jsteel2

So the argument is that R can only be an equivalence relation if the set A has atleast two elements?

The question is "Can R be an equivalence relation on A?" so if the answer is yes, you only need to find one example then you're done. I believe the one-element set is the simplest example... for example A = {1} and R = {A,A,{(1,1)}}.