Determine whether each of these functions from Z to Z is one-to-one.

Answer:Step-1: In this problem we need to determine whether each of the given functions from Z to Z is one-to -one (or) not. One -to-one function: A function for which every element of the range of the function corresponds to exactly one element of the domain. Test for one -to-one functions : If f(a) = f(b) implies that a = b , then f is one-to-one.1. In this problem we need to determine f(n) = n-1 is one -to-one (or) not. Let us consider , , where . Now , test for one-to-one functions: Assume that , f(n) = f(p) , since given f(n) = n-1 . Therefore , f is one -to-one function.Step-2:b) In this problem we need to determine is one -to-one (or) not. Let us consider , , where . Now , test for one-to-one functions: Assume that , f(m) = f(n) , since given . By the definition of one-to-one function the given function f is not one-to-one function.Because n has two values. Therefore , f is not one-to-one function.Step-3: c) In this problem we need to determine is one -to-one (or) not. Let us consider , , where . Now , test for one-to-one functions: Assume that , f(r) = f(n) , since given By the definition of one-to-one function the given function f is one-to-one function.Because n has exactly one value. Therefore , f is one-to-one function.Step-4: d) In this problem we need to determine is one -to-one (or) not. The given function f is not one-to-one .Because if n= 3 then , since n = 3. = 2. If n= 4 then , since n =4 = 2 Hence , if n= 3 and 4 the value of f(n) = 2.So , by the definition of one to one function f is not one-to-one function.NOTE: The symbols for floor and ceiling are like the square brackets [ ] with the top or bottom part missing. Example : Therefore , f is not one -to-one function.