Let A be a set with |A| = n, and consider the order for the listing of its elements as fixed. For R ⊆ A × A, let M (R) denote the corresponding relation matrix.
(a) Prove that M (R) = 0 (the n × n matrix of all 0's) if and only if R = ∅.
(b) Use the result of Exercise 11, along with the Principle of Mathematical Induction, to prove that M(Rm) = [M(R)]m, for all m ∈ Z+.