1. Let X = {1, 2,3, 4). Define G : X + X as follows: For each s E X, G(s) = the number t in X such that (at) mod 5 = 1. Draw the arrow diagram for G. Is G well-defined? Is G a function? Justify your answer. 2. Let A = {1, 2,3, 4, 5} and define a function F : (A) - Z as follows: For all sets X in (A), F(X) = the number of elements in (A - X) (a) Find F(0), F({2,3,5)), and F(A). (b) Find the preimage of 1. (c) What is the range of F? 3. Let S be the set of all strings of a's and b's. Define a function f : S - Z as follows: For each string s in S, 1 (s ) = the number of b's to the left of the left-most a in s 0 if s contains no a's. Find f(ababa), f(bbbabab), f(bb), and f(bbabba). What is the range of f? 4. Determine whether each of the following functions is one-to-one or onto or both or neither. Explain your answer. (a) The function f : R -> Z defined by f(x) = [x - 1]. (b) Let X = {0, 1, 2,3, 4, 5} and define a function g : X -+ X by g(x) = 31 mod 6. (c) The function H : Z x Z -+ Z defined by H(m, n) = (2m) . n. 5. Determine whether the following function is one-to-one or onto or bijective or neither. Find an inverse function if it exists. Justify your answer. (a) Define a function f : IR- (1) - R - (1} by f(a) - (b) Define a function 9 : R - {{} - R by 9(x) = 1 - 2.c 5x - 1