Math 325K. Fall 2015 Guide for the Final Exam Prof. Hector E. Lomeli Study the following documents Guide for midterm 2. Additional Problems 3. Guide for midterm 3. Additional Problems 4. Concentrate on the following problems in chapter 8. 8.2: 30. 8.3: 22, 23. 8.4: 11-16. Let J41 = {0, 1, . . . , 40}. Let F : J41 J41 be the function dened by F (x) = 3x + 2 mod 41, Find two integers a, b Z such that, if we let G : J41 J41 be the function dened by G(x) = a x + b mod 41, then F G = I and G F = I. Let R be the relation dened on Z as follows: x R y 8|x2 y 2 . Prove the following. a) R is an equivalence relation. b) There are only three equivalence classes: [0], [1], [2]. Let Q be the relation dened on Z as follows: x Q y 15|x4 y 4 . Prove the following. a) Q is an equivalence relation. b) If 3 | x and 5 | x then xQ0. c) If 3 | x and 5 | x then xQ1. d) If 3 | x and 5 | x then xQ6. e) If 3 | x and 5 | x then xQ10. f ) There are only four equivalence classes: [0], [1], [6], [10]