Exercise 4.2.4: Properties of functions on strings and power sets

Exercise 4.2.4: Properties of functions on strings and power sets. 0 About 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. o f: {0, 118 + {0, 1). The output off is obtained by taking the input string and replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) = 110 f:{0, 113+{0, 133. The output off is obtained by taking the input string and reversing the bits. For example f(011) = 110. (0) f:{0, 133+ {0,1}4. The output off is obtained by taking the input string and adding an extra copy of the first bit to the end of the string. For example, f(100) = 1001. (9) Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8} and let B = {1}. f: P(A) + P(A). For X S A, f(x) = X-B. Recall that for a finite set A, P(A) denotes the power set of A which is the set of all subsets of A. Exercise 4.2.4: Properties of functions on strings and power sets. 0 About 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. o f: {0, 118 + {0, 1). The output off is obtained by taking the input string and replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) = 110 f:{0, 113+{0, 133. The output off is obtained by taking the input string and reversing the bits. For example f(011) = 110. (0) f:{0, 133+ {0,1}4. The output off is obtained by taking the input string and adding an extra copy of the first bit to the end of the string. For example, f(100) = 1001. (9) Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8} and let B = {1}. f: P(A) + P(A). For X S A, f(x) = X-B. Recall that for a finite set A, P(A) denotes the power set of A which is the set of all subsets of A