Question 1. (a) Give the definition of A≈ B; i.e. the set A is equinumerous to the set B. (b) Prove that if A is any set, then AP(A). Question 2. (a) State the Schröder-Bernstein Theorem. (b) Using the Schröder-Bernstein Theorem, prove that N × P(N) ≈ P(N). (c) A function f : N → N is a derangement of N if f is a bijection such that f(n) n for all n € N. Determine whether the set Der(N) of derangements of N is countable or uncountable. em Hint: consider the bijections : N N such that for all nЄ N, Question 3. [{3n, 3n+1, 3n+2}] = {3n, 3n+1, 3n+2}. (a) Define the addition operation + λ for cardinal numbers к and X. (b) Prove that No +2 No .סא2 (c) Define the multiplication operation к A for cardinal numbers к and A. (d) Prove that if K, A and 0 are cardinal numbers, then кk Question 4. (a) Give the definition of a well-ordering < of a set A. . (b) Let be the linear ordering on N × N defined by (a, b) < (c, d) if either: ⚫ a < c; or ac and b < d. Prove that is a well-ordering of N × N. Note: You do not need to prove that is a linear ordering of N × N. (c) Determine whether (N× N, <)≈ (N,<).