Question

The function $f : \mathbb{N} - \\{1\\} \to \mathbb{N}$; defined by $f(n)$ = the highest prime factor of $n$, is:

• A. both one-one and onto
• B. one-one only
• C. onto only
• D. neither one-one nor onto

Answer 1

Answer: (D)

neither one-one nor onto

Answer 2

Step 1. Understanding the Function $f(n)$: The function $f(n)$ maps each natural number $n$ (excluding 1) to its highest prime factor. For example:

$f(10) = 5, \quad f(15) = 5, \quad f(18) = 3$

Step 2. Checking if $f(n)$ is One-One: For a function to be one-one (injective), each distinct input must map to a distinct output. However, different values of $n$ can have the same highest prime factor. For instance:

$f(10) = f(15) = 5$

- Since different numbers can yield the same highest prime factor, $f(n)$ is not one-one.

Step 3. Checking if $f(n)$ is Onto: For $f(n)$ to be onto (surjective), every natural number should appear as an output of $f(n)$. However, not all natural numbers are prime. Since $f(n)$ only outputs prime numbers, it cannot cover all natural numbers. Therefore, $f(n)$ is not onto.

Since $f(n)$ is neither one-one nor onto, the correct answer is $(4)$.