Question

Determine which of the following functions are one-to-one.

(a) f1:{1, 2, 3, 4, 5} + {a,b,c,d}; fi(1) = b, f1(2) = c, fi(3) = a, f1(4) = a, f1(5) = c
(b) f2:{1,2,3,4} + {a,b,c,d, e}; f2(1) = C, f2(2) = b, f(3) = a, f2(4) = d
(c) f3:ZZ; fs(n) = -n 2n if n < 0
(d) $4: Z+Z; f(n) = -3n if n > 0

Answer 1

Function (a) is not one-to-one because it maps different inputs to the same output. Function (b) is one-to-one since each input has a unique output, and function (d) is also one-to-one as it maps positive integers to unique negative integers. Function (c) cannot be determined from the information provided.

Step-by-step explanation:

The functions that are one-to-one are those in which each element of the domain is mapped to a unique element of the codomain, meaning no two different elements in the domain map to the same element in the codomain.

For function (a), f1, we observe that f1(3) = a and f1(4) = a. Since two different inputs give the same output, this function is not one-to-one.

For function (b), f2, each input maps to a distinct output, making this function one-to-one.

Function (c), f3, is described as fs(n) = -n 2n if n < 0. Without a clear definition of the function for the full domain of integers (ZZ), we cannot determine if it is one-to-one.

Function (d), which uses the set of positive integers Z+ and the expression -3n if n > 0, is a one-to-one function, as each positive integer n will have a unique negative integer as its image.