Question

For each of the following functions, indicate whether the function has a well-defined inverse: If the inverse is well-defined, give the input/output relationship of f−¹

(a) f: Z → Z f(x) = x+3
(b) f: Z → Z,f(x) = 2x+34
(c) f: R → R f(x) = 2x+3
(d) Let A be defined to be the set {1,2,3,4,5,6,7,8} f: P(A) → {0,1,2,3,4,5,6,7,8} For X⊆A,f(X)=∣X∣. Recall that for a finite set A,P(A) denotes the power set of A which is the set of all subsets of A.
(e) Let A be defined to be the set {1,2,3,4,5,6,7,8}f: P(A) → P(A). For X⊆A,f(X)=A−X. Recall that for a finite set A. P(A) denotes the power set of A which is the set of all subsets of A.
(f) f:{0,1}³ →{0,1}³.The output of f 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.
(g) f:{0,1}³ →{0,1}³.The output of f is obtained by taking the input string and reversing the bits. For example, f(011)=110.
(h) f:{0,1}³ →{0,1}³. The output of f is obtained by taking the input string x, removing the first bit of x, and adding the bit to the end of x. For example, f(011)=110.
(i) f: Z × Z → Z × Z,f(x,y) = (x+5,y−2)
(j) f: Z × Z → Z × Z,f(x,y)=(1−y,x)

Answer 1

For some of the given functions, their inverses are well-defined, while for others they are not. The inverse of a function undoes the original function and maps the output back to the input. The process of finding the inverse involves solving equations and manipulating the variables.

Step-by-step explanation:

(a) The function f(x) = x+3 has a well-defined inverse. To find the inverse, we need to solve the equation y = x+3 for x. By subtracting 3 from both sides, we get x = y-3. So the inverse of f(x) is f^(-1)(x) = x-3.

(b) The function f(x) = 2x+34 has a well-defined inverse. We need to solve the equation y = 2x+34 for x. By subtracting 34 from both sides, we get 2x = y-34. Dividing both sides by 2, we get x = (y-34)/2. So the inverse of f(x) is f^(-1)(x) = (x-34)/2.

(c) The function f(x) = 2x+3 has a well-defined inverse. We need to solve the equation y = 2x+3 for x. By subtracting 3 from both sides, we get 2x = y-3. Dividing both sides by 2, we get x = (y-3)/2. So the inverse of f(x) is f^(-1)(x) = (x-3)/2.

(d) The function f(X) = |X| does not have a well-defined inverse. This is because the power set of a set A includes all possible subsets of A, and the cardinality of those subsets is not uniquely determined by f(X). For example, f({1,2}) = 2, but f({1,2,3}) = 3.

(e) The function f(X) = A-X does have a well-defined inverse. The inverse of f(X) is f^(-1)(X) = A-X, since applying f(X) twice will give us back the original set X.

(f) The function f(x) replaces the first bit of the input string with 1, regardless of whether it is 0 or 1. So, the inverse of f(x) is f^(-1)(x) = 0 + x[2:4], where x[2:4] represents the last 3 bits of the input string.

(g) The function f(x) reverses the bits of the input string. So, the inverse of f(x) is f^(-1)(x) = x[3] + x[2] + x[1], where x[3], x[2], and x[1] represent the 3 bits of the input string in reverse order.

(h) The function f(x) removes the first bit of the input string and adds it to the end. So, the inverse of f(x) is f^(-1)(x) = x[2:3] + x[1], where x[2:3] represents the last 2 bits of the input string and x[1] represents the first bit.

(i) The function f(x,y) = (x+5, y-2) has a well-defined inverse. To find the inverse, we need to solve the equation (a,b) = (x+5, y-2) for x and y. By subtracting 5 from the first component and adding 2 to the second component, we get x = a-5 and y = b+2. So the inverse of f(x,y) is f^(-1)(x,y) = (x-5, y+2).

(j) The function f(x,y) = (1-y, x) has a well-defined inverse. To find the inverse, we need to solve the equation (a,b) = (1-y, x) for x and y. By subtracting b from 1 and swapping the variables, we get x = b and y = 1-a. So the inverse of f(x,y) is f^(-1)(x,y) = (1-y, x).