4. [14 marks] Working with functions. In this question, we will explore various properties of functions. You may want to review the basic denitions and terminology introduced on pages 15716 of the course notes. Then, read the following denitions carefully. Denition: A function f : A > B is onetoone iff no two elements of A have the same image. Symbol ically, Vahog E A,f(a1) : f(a2) :> 0.1 : a2. (3) Denition: A function f : A a B is onto ijf every element of B is the image of at least one element from A. Symbolically, VI) 6 B, 3a E A,f(o) = b. (4) Denition: For all functions f : A > B and g : B ) 0', their composition is the function g o f : A :> C dened by: Va 6 A=(90f)(a) =9(f(a))' (5) (a) [4 marks] i. Prove that 91 : Z a Z; g1(:1:) : r i 4 is both one-to-one and onto. ii. Prove that 9;; : R > R; 92(r) = II] -l a: is neither onetoone nor onto. (b) [4 marks] Give explicit, concrete denitions for two functions f1, f2 : Z > Z+ such that: i. f2 is onto but not onetoone, ii. fl is onetoone but not onto, and prove that each of your functions has the desired properties. (C) [6 marks] Let f z A > B and g : B ) C be arbitrary functions. Prove or disprove each of the following. In each case, rst write down in symbolic notation the exact statement you are attempting to prove (either the original statement or its negation). i. If g o f is one-to-one, then f is also one-toone. ii. If g o f is onto, then 9 is also onto. iii. If g o f is both one-toone and onto, then f and g are also both onetoone and onto