1. [6 points] Prove or disprove each statement. [hint: see section 4.2 under additional problems] a. If n is an odd integer, then 131 =n-1 b. For any real number x, [[ x] ]=[ x] . 2. [6 points] For each of the functions below, indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why. [hint: see section 4.3 under additional problems] a. f : ZxZ - ZxZ,f ( x, y)= 1=1,y-2 b. f : Z X Z - ZxZ, f ( x, y )= /=1, 5y-2 c. f :ZXZ - ZxZ,f (x,y)=(1-y,1-x 3. [6 Points] Indicate which of the following functions have inverses for the domain X and target Y given below, and give the inverses: X = {a, b, c, d}, Y = {1, 2, 3, 4} a) f1 = {(a, 3), (b, 1), (c, 2), (d, 2) } b) f2 = {(a, 4), (b, 2), (c, 3), (d, 1) } c) f3 = {(a, 3), (b, 4), (c, 2), (d, 4) } 4. [6 Points] Consider functions f, g, h, defined below, all of which have R as their domain and target. f ( x) = 2x + 1, g(x) = x2, h(x) =x+3 Compute the following: a) (fog) (2) b) (go f) (2) c) (hog of) (0) 5. [6 points] For each expression, give an equivalent expression that does not use the log function. [hint: see section 4.6 under additional problems]