Are Injective Functions Also Surjective: Exploring the Relationship Between Injectivity and Surjectivity

Introduction

In the realm of mathematics, functions are fundamental concepts with diverse properties that help us understand the relationships between sets. Two such important properties are injectivity and surjectivity. While these terms may sound similar, they describe distinct characteristics of functions. This article delves into the differences between these properties and explores the conditions under which they coincide.

Injective Functions: The One-to-One Property

A function (f: A to B) is injective, or one-to-one, if every element in set (A) is mapped to a unique element in set (B). Formally, if for all (x_1, x_2 in A), (f(x_1) f(x_2)) implies (x_1 x_2), then (f) is injective. This means that no two elements in the domain map to the same element in the codomain.

Example of an Injective Function

Consider the function (f: {1, 2} to {a, b, c}) defined by (f(1) a, f(2) b). This function is injective because each input maps to a unique output. However, it is not surjective because the element (c) does not have a corresponding input in (A).

Surjective Functions: The Onto Property

On the other hand, a function (f: A to B) is surjective, or onto, if every element in set (B) is the image of at least one element in set (A). Formally, for every (b in B), there exists at least one (a in A) such that (f(a) b). This means that every element in the codomain is mapped to by at least one element in the domain.

Example of a Surjective Function

Consider the function (g: {1, 2} to {a, b}) defined by (g(1) a, g(2) a). This function is surjective because both elements in the codomain (a) and (b) have preimages in the domain. However, it is not injective because both inputs map to the same output.

Relationship Between Injective and Surjective Functions

Injective Functions Are Not Necessarily Surjective

A function can be injective without being surjective. For instance, the function (f: {1, 2} to {a, b, c}) defined by (f(1) a, f(2) b) is injective but not surjective because there is no input that maps to (c).

Surjective Functions Are Not Necessarily Injective

Similarly, a function can be surjective without being injective. Consider the function (g: {1, 2} to {a, b}) defined by (g(1) a, g(2) a). This function is surjective because every element in the codomain is mapped to by at least one element in the domain, but it is not injective because both inputs map to the same output.

Bijective Functions: The Perfect Match

A function that is both injective and surjective is known as a bijective function or a one-to-one correspondence. Bijective functions are particularly interesting because they have an inverse function. For example, consider the function (y f(x) x^2) on the domain (D_f mathbb{R}). This function is both injective and surjective, with the range (R_f [0, infty)). Thus, there exists an inverse function (x g(y) sqrt{y}) on the domain (D_g [0, infty)) with the range (R_g D_f mathbb{R}).

Why Domain Definition Matters

It is crucial to note that the domain is a fundamental part of a function's definition. Two functions are considered different if they differ in their domains. For example, the function (f: mathbb{R} to mathbb{R}, f(x) e^x) is injective but not surjective, while the function (g: mathbb{R} to (0, infty), g(x) e^x) is both injective and surjective. This highlights the importance of domain specification in understanding the properties of functions.

Finite Domains and Codomains

In the case of finite domains and codomains, a function can be both surjective and injective. For instance, consider the function (f: mathbb{Z} to mathbb{Z}, f(n) 2n). This function is neither injective nor surjective on the set of integers. However, if the domain and codomain are both the set of even integers, then the function is bijective and has an inverse.

Conclusion

In conclusion, injective functions are not necessarily surjective and vice versa. The coincidence of these properties depends on the specific domain and codomain of the function. Understanding the distinction between injective and surjective functions is crucial for grasping the properties of various mathematical functions and their applications in different fields.