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

The Correct Option is D

To determine the nature of the function \( f(n) \), which is defined as the highest prime factor of \( n \) for \( n \in \mathbb{N} - \{1\} \) (i.e., natural numbers excluding 1), we need to analyze its properties.

Step-by-Step Analysis 

1. Understanding the Function

The function \( f(n) \) takes a natural number \( n \) and returns the highest prime factor of \( n \). For example, \( f(10) = 5 \) because the prime factors of 10 are 2 and 5, with 5 being the highest.

2. Checking if the Function is One-One (Injective)

A function is one-one if each element of the domain maps to a distinct element in the codomain. Consider:

• \( f(6) = 3 \) because the prime factors of 6 are 2 and 3.
• \( f(9) = 3 \) because the prime factor of 9 is 3 (since \( 9 = 3 \times 3 \)).

Both 6 and 9 give the same highest prime factor (3), which shows that \( f \) cannot be one-one because two different numbers in the domain have the same image.

3. Checking if the Function is Onto (Surjective)

A function is onto if every element of the codomain has a preimage in the domain. The codomain is \(\mathbb{N}\), but for \( f(n) \), the possible outputs are only primes.

• For instance, the number 4 is in \(\mathbb{N}\), but there is no natural number \( n \) (other than 1) such that \( f(n) = 4 \) because 4 is not a prime number.

This shows that not all natural numbers can be represented as the highest prime factor of any \( n \), proving that the function is not onto.

Conclusion

Since \( f(n) \) is neither one-one (injective) because it maps different numbers to the same highest prime factor, nor onto (surjective) because not all elements in the codomain (natural numbers) are covered as outputs, the correct answer is:

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 \( (4) \).