Hello, how to do questions 3 and 4? Thank you

Question 3. [total 11 marks] Let U be a countable infinite set. Recall that if A C U, then P(A) = {SCU : SC A) = {SC U : S\\ A is empty}. For each subset A C U, define Po (A) = {S C U : S S A and S is finite} and Py(A) = {SC U : S \\ A is finite]. Consider a fixed A C U. The aim of this question is to determine when Py (A) is countable. (a) Suppose U = Zy, and A = (2k : k E Z,,]. Is S E Py (A) in the three cases below? (i) S = (3k : KEZ,.]. (ii) S = XUY, where X = (4k : k EZ>} C A and Y = (4k + 1 :KEZ>,} SU\\A. (iii) S = X UY, where X = {6k : k E Z>(} S A and Y = {1,3,5, ...,1231} C U \\ A. No justification is needed. [3 marks] (b) Convince yourself that P(A) S Pi (A). [0 mark] (c) Convince yourself that P; (A) = {XUY : X E P(A) and Y E P. (U \\ A)]. [0 mark] (d) Define f: P(A) X Pu (U \\ A) - Pu(A) by setting, for all X E P(A) and all Y E Pu (U \\ A), f ( X , Y ) = XUY. Using (c), prove that f is a bijection, and thus | P(A) x Pw (U \\ A)I = [Pi(A)I. [3 marks] (e) Use (d) to prove that if A is finite, then Pi(A) is countable. [3 marks] (f) Use (b) to prove that if A is infinite, then Pi(A) is uncountable. [2 marks] In addition to the results proved in the notes, the exercises, and the tutorials, you may quote the following facts without proof. (A) For all sets A and S, if S C A, then S \\ A = 0. (B) For all sets A, S and U, if S C U, then S \\ A C U \\ A. (C) For all sets A and S, S = (SnA) U (S \\ A). (D) For all sets U, X and Y, if X C U and Y C U, then XUY C U. (E) For all sets A, U, X and Y, if X C A and Y C U \\ A, then (XU Y) \\ A = Y. CS Scanned with CamScanner(d) Define f: P(A) x Pw (U \\ A) - Pu(A) by setting, for all X E P(A) and all Y E Pw (U \\ A), f ( X, Y) = XUY. Using (c), prove that f is a bijection, and thus |P(A) X P. (U \\ A)| = [P;(A)I. [3 marks] (e) Use (d) to prove that if A is finite, then Py (A) is countable. [3 marks] (f) Use (b) to prove that if A is infinite, then Py (A) is uncountable. [2 marks] In addition to the results proved in the notes, the exercises, and the tutorials, you may quote the following facts without proof. (A) For all sets A and S, if S C A, then S \\ A = 0. (B) For all sets A, S and U, if S C U, then S \\ A CU \\ A. (C) For all sets A and S, S = (Sn A) U (S \\ A). (D) For all sets U, X and Y, if X C U and Y C U, then XU Y S U. (E) For all sets A, U, X and Y, if X C A and Y C U \\ A, then (XU Y) \\ A = Y. (F) For all sets A and U, for all X1,X2 C A and all Y1, Yz C U \\ A, if X1 U Y1 = X2 U Y2, then X1 = X2 and Y1 = Yz. (G) For all sets A and U, U \\ ACU. (H) (Corollary of Theorem 10.4.2.) If P and Q are countable sets, then P X Q is countable. (1) (Corollary of Tutorial 8 Question 7(b).) If B is a countable set, then Pw (B) is countable. Question 4. [2 marks] Given a piece of graph paper. Starting at the origin (0,0), draw a path to (10,10) that stays on the grid lines and goes only up and right (e.g. (0,0) - (0,1) - (2,1) - (2,10) - (10,10)). How many such paths are possible? Explain in one sentence how your answer is obtained. CS Scanned with CamScanner