Relations can be composed just as functions are composed (for instance ).. note that the "o" is meant to be the "of" symbol but didn't know how to do it in latex.
For each of the following below, either prove or provide a counterexample:
a.) If is reflexive and if is reflexive, then is reflexive.
b.) If is symmetric and is symmetric, then is symmetric.
c.) If is transitive and is transitive, then is transitive.
I'm assuming that, based on your results, still wouldn't be reflexive.
For b, showing if its symmetric,
Suppose . Also, would also equal that if they're both symmetrical. So then the composition of R and S would be symmetrical?
And for c, transitivity is a little harder since we know if A = B, and B = C, then A = C. So I'd have to set up 3 different subsets? Perhaps drawing digraphs would help?