Let C and S be fixed sets where C is a subset of S. Let R be a relation defined on S by (A,B) belong to R if and only if A intersect C = B intersect C. Determine whether R is an equivalence relation.
"A relation on S" is a subset of S x S. Here, A and B are subsets of S rather than elements of S; therefore, R is a relation on the powerset of S.
Suppose that R' is a relation on some set X and suppose that there exists a function f such that that (x, y) ∈ R' iff f(x) = f(y). Then R' is an equivalence relation.