Animated Solution for Mathematics - Functions: A function $f$ from the set of natural numbers to integers defined by $f(n)=\{\begin{array}{ll}\frac{n-1}{2},& \text{when }n\text{ is odd}\\ -\frac{n}{2},& \text{when }n\text{ is even}\end{array}$ is

Imagine you are standing at the edge of two vast mathematical landscapes. On one side, we have the natural numbers $\mathbb{N}=\{1,2,3,\dots \}$, the counting numbers that form the bedrock of our arithmetic.

On the other side, we have the integers $\mathbb{Z}=\{0,1,-1,2,-2,\dots \}$, a world that extends infinitely in both positive and negative directions. Our task is to understand the bridge between them, defined by the function $f(n)$.

This function is not a simple, uniform rule; it is a piecewise function, a 'split personality' that behaves differently depending on the nature of the input $n$. The function is defined as:

Let's test this with the first few natural numbers to observe the pattern:

For $n=1$ (odd):

For $n=2$ (even):

For $n=3$ (odd):

For $n=4$ (even):

A function is one-one, or injective, if every input maps to a unique output. Here, the odd inputs map to the non-negative integers $\{0,1,2,\dots \}$, while the even inputs map to the negative integers $\{-1,-2,-3,\dots \}$.

Since these two sets of outputs are disjoint—a number cannot be both non-negative and negative—no two different inputs can ever map to the same output. Thus, the function is one-one.

A function is onto, or surjective, if its range covers the entire codomain. Our range is the union of the non-negative integers and the negative integers, which is exactly the set of all integers $\mathbb{Z}$.

Since the range equals the codomain, the function is onto.

We have shown that the function is both one-one and onto, making it a bijection. This elegant piecewise construction perfectly maps the natural numbers to the integers, a fundamental concept in set theory and function analysis.