# Equivalence relation and total ordering problem

• Jun 26th 2010, 11:21 PM
jsteel2
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?
• Jun 26th 2010, 11:36 PM
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Quote:

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.
• Jun 27th 2010, 05:25 PM
jsteel2
So the argument is that R can only be an equivalence relation if the set A has atleast two elements?
• Jun 27th 2010, 05:48 PM
undefined
Quote:

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)}}.