Note: To prove a statement is false, you must write out the negation of the statement and prove that. Questions: 1. Prove or disprove each of the following statements. (a) For all functions f, g and h from Z to Z, if go f = go h then f = h. (b) For all functions f, g and h from Z to Z, if fog=hog then f=h. (c) For all functions f, g and h from Z to Z, if g is one-to-one and go f = go h then f = h. (d) For all functions f, g and h from Z to Z, if g is onto and fog=hog then f = h. 2, Let f:Z Z be the function defined by f (z) = 2x 3 for each # Z. Recall that I3 is the identity function on Z defined by Iz (z) = z for all x Z. (a) Is f one-to-one? Prove your answer. (b) Is f onto? Prove your answer. (c) Is there a function g : Z Z so that f o g = Iz7 Prove your answer. (d) Is there a function h : Z Z so that ho f = I3? Prove your answer. 3. Let A={1,2,3,4}. Let F be the set of all functions from A to A. Let R be the relation on F defined by: For all f,g F, fRg if and only if fog(1) =2. (a) Is R reflexive, symmetric, antisymmetric, transitive? Prove your answers. (b) Is it true that for all functions f F, there exists a function g F so that fRg? Prove your answer. (c) Is it true that there exists a function f F so that for all functions g F', fRg? Prove your answer. (d) How many functions f F are there so that [y Rf? Explain