Let A be a nonempty set and fix the set B, where B A. Define the

Question:

Let A be a nonempty set and fix the set B, where B ⊆ A. Define the relation R on P(A) by X R Y, for X, Y ⊆ A, if B ⋂ X = B ⋂ Y.
(a) Verify that R is an equivalence relation on P(A).
(b) If A = {1, 2, 3} and B = {1, 2}, find the partition of P(A) induced by R.
(c) If A = {1, 2, 3, 4, 5} and B = {1, 2, 3}, find [X] if X = {1, 3, 5}.
(d) For A = {1, 2, 3, 4, 5} and B = {1, 2, 3}, how many equivalence classes are in the partition induced by R?