11. In each of the following, find the integer m that satisfies the given condition and such that 0

11. In each of the following, find the integer m that satisfies the given condition and such that O < m < 11. (a) [4397] = [m] in Z12 (b) [4397] = [m] in Z12 (c) [8] [m] = [O] in Z12 (d) [5] @ [m] = [1] in Z12 12. For (a, b) and (c, d) in R2, define (a, b) (c, d) to mean that 3a 2b = 3c 2d. You are given that is an equivalence relation on R2. Use a theorem from the notes (one that gives four equivalent statements) to verify each of the following. 13. For (81, Yl) and ($2, Y2) in R x R define (81, Yl) to mean that Yl = x}. You are given that is an equivalence relation on IR x IR. E denotes the set of all equivalence classes of R x R for the equivalence relation . (a) Give a set description of the equivalence class [(0, O)] and then graph the equivalence class in R x R. (b) Show that [(1, 2)] = 1)] and [(2, 6)] (c) Prove that for every [(a, b)] e E there exists c e R such that 14. Define a binary operation * on the set Z of integers by a * b = ab a b + 2 for all a, b e Z. Prove there exists an integer e such that for every integer a we have a * e = a. 15. Prove by induction that (1.1)n 1 + for every integer n 1. (Hint: It may 1 help to write 1.1 = 1 -4- 10