Answer to Question #314588 in Discrete Mathematics for bhupendra

Question

Determine whether each of the function from 𝑍 to Z is one to one

(a) 𝑓(𝑛) = 𝑛 − 1

(b) 𝑓(𝑛) = 𝑛² + 1

Answer 1

(a) For any $m\in\mathbb{Z}$ the equation $f(n)=n-1=m$ has a unique solution: $n=m+1$. This means that the function $f(n)$ is surjective and injective (one-to-one), and it's inverse is $f^{-1}(m)=m+1$ .

(b) For any $n\in\mathbb{Z}$ $f(n)=f(-n)=n^2+1$. Thus, different values of argument may give the same result. Therefore, the function $f(n)$ is not one-to-one.

Answer. (a) yes; (b) no.