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September 26th, 2014, 04:04 AM   #1
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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

~ ∩ ({x ∈ C: x~x} × {x ∈ C: x~x}) is a equivalence relation on {x ∈ C: x~x}.

How should I interpret the intersection of two sets of ordered pairs generated by ~ and the cartesian product?
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September 26th, 2014, 02:59 PM   #2
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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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