Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and T = {2, 4, 6, 8}. Let R be the relation on P (S), the power set of S, defined by: For all A,B = P (S), (A,B) = R if and only if A U T = B U T.

 (a) Prove that R is an equivalence relation on P (S). (

b) How many equivalence classes are there? Explain.

 (c) How many elements does [{1, 2}], the equivalence class of {1, 2}, have? Explain. 

(d) How many subsets X = P (S) are there so that |X| = 5 and X is not an element of [{1, 2}]? Explain.