Bijection from Natural Numbers to Square Numbers

Show that a map from the natural numbers to the square numbers, defined by mapping $x$ to $x^2$, is bijective.

In this problem, we examine a fundamental concept in real analysis by determining whether a given function is bijective. Bijectivity is a property of functions where each element in the domain is mapped to a unique element in the codomain, and vice versa, meaning that both the injective (one-to-one) and surjective (onto) conditions are satisfied. The function you are given maps each natural number $x$ to its square $x^2$.

To show that this function is bijective, you need to understand both the concepts of injection and surjection. For injection, focus on proving that if two outputs of the function are equal, their corresponding inputs must also be equal. Mathematically, if $x^2 = y^2$, then $x$ must equal $y$ for natural numbers $x$ and $y$. This is because natural numbers are positive, so the square root is a well-defined operation.

For surjection, demonstrate that for every square number in the codomain, there exists a natural number that maps to it. Since every natural number $n$ has a square $n^2$ in the codomain and this function results in the set of all square numbers, it satisfies the onto condition. By concluding both these aspects, you establish that the mapping is bijective, underscoring the critical nature of functions in understanding mathematical structures and relationships.