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Im stuck with the following question:
Q: R is a relation on X. Prove if R is symmetric, then Rt is symmetric (Note Rt is the transitive closure of R).
I know that for R to be symmetric, we need xRy and yRx. Also, Rt is the smallest transitive relation containing so Rt is a subset of R.
So in thinking about this, I think I need to show that assuming R is symmetric, that 1) R is a subset of Rt^-1 and that Rt^-1 is transitive. Is this correct? If so, how do I prove 1)?
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You need to look at exactly what relations you will be adding to R. Specifically, if a R b and b R c then you need to add a relation. However, if that's the case, then b R a and c R b because R is symmetric. Thus, c R b and b R a, so you need to also add in c R a if it isn't there.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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