please answer all the questions.

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=ho g then f =h. 2, Let f:Z Z be the function defined by f (x) = 2z 3 for each x Z. Recall that Iz is the identity function on Z defined by Iz (x) = z for all z 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 = Iz? Prove your answer. (d) Is there a function h : Z Z so that he f = I;7 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(l) =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 [4Rf? Explain