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

We define a function $f:\mathbb{N}-\{1\}\to \mathbb{N}$ such that $f(n)$ is the highest prime factor of $n$. The domain is $\{2,3,4,5,6,\dots \}$, excluding $1$ because $1$ lacks prime factors.

The codomain is the set of all natural numbers $\mathbb{N}=\{1,2,3,4,5,\dots \}$. Our objective is to determine if this function is one-one (injective) and onto (surjective).

A function is one-one if every distinct input maps to a distinct output. Let us test this by evaluating specific inputs:

For $n=2$, the prime factorization is $2$, so $f(2)=2$.

For $n=4$, the prime factorization is $2^2$, so the highest prime factor is $2$. Thus, $f(4)=2$.

Since $f(2)=f(4)$ but $2 eq 4$, we have identified a collision. Therefore, the function is not one-one; it is a many-one function.

A function is onto if every element in the codomain is covered by at least one element from the domain. Let us examine the codomain $\mathbb{N}=\{1,2,3,4,5,\dots \}$.

The smallest prime number is $2$. Because our machine only outputs prime numbers, the value $1$ in the codomain can never be reached.

Furthermore, composite numbers like $4$ or $6$ are also unreachable as outputs. The range of the function is the set of all prime numbers:

Since the range is a proper subset of the codomain, there are infinitely many numbers in the codomain that are never mapped to. Thus, the function is not onto; it is an into function.

Through our analysis, we have determined that the function $f(n)$ is neither one-one nor onto.

This function acts as a probe into the "DNA" of numbers, extracting the largest prime component. As demonstrated by $f(3)=3$ and $f(6)=3$, the mapping confirms the many-one nature of the relationship.

By testing simple cases, we have successfully unraveled the behavior of this function across the entire set of natural numbers.