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September 26th, 2014, 03:04 AM  #1 
Newbie Joined: Sep 2014 From: Norway Posts: 2 Thanks: 0  Equivalence relation on cartesian product
I need some help to interpret the statement below: Let ~ be a relation on a set C. ~ is a transitive, reflexive and symmetric relation on C. Show that ~ âˆ© ({x âˆˆ C: x~x} Ã— {x âˆˆ C: x~x}) is a equivalence relation on {x âˆˆ C: x~x}. How should I interpret the intersection of two sets of ordered pairs generated by ~ and the cartesian product? 
September 26th, 2014, 01:59 PM  #2 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 
If $\sim$ if reflexive, as the problem says, then $\{x\in C: x\sim x\}=C$ and ${\sim}\cap(C\times C)={\sim}$. I think it makes more sense not to assume that $\sim$ is reflexive. Then it is a PER, and it is an equivalence relation on the set where it is reflexive.


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