Understanding Surjective and Injective Functions
In the realm of mathematical functions, surjective (onto) and injective (one-to-one) functions are fundamental concepts that help us understand the relationship between the domain and codomain of a function. A surjective function ensures that every element in the codomain is the image of at least one element in the domain, while an injective function guarantees that each element in the domain maps to a unique element in the codomain. However, a function can be surjective without being injective, and vice versa. This article will explore several examples of functions that are exclusively surjective but not injective.
Linear Function: An Example of a Surjective but Not Injective Function
Definition of the Function:
Let f: R → R be defined by f(x) x2.
Explanation:
This function is not surjective because it does not cover negative numbers in the codomain. Therefore, to make it surjective, we can restrict the codomain to [0, ∞). The function g(x) x2 with codomain R is surjective but not injective. For example, both g(1) 1 and g(-1) 1 map the same output, demonstrating that it is not injective.
Modulo Function: Another Example of a Surjective but Not Injective Function
Definition of the Function:
Let f: Z → Z/3Z be defined by f(x) x mod 3.
Explanation:
This function is surjective because every equivalence class 0, 1, 2 in Z/3Z has at least one pre-image in Z. However, it is not injective because multiple integers map to the same class e.g., f(0) 0, f(3) 0, f(-3) 0. This shows that the function is not one-to-one.
Constant Function: A Basic Example of a Surjective but Not Injective Function
Definition of the Function:
Let f: R → R be defined by f(x) c, where c is a constant.
Explanation:
This function is not injective because every input maps to the same output c. However, if the codomain is restricted to {c}, such as in the example f(x) 1 mapping from R to {1}, the function is then surjective. Despite being surjective, it remains non-injective because both 1 and -1 get mapped to the same value c.
Quadratic Function: A Further Illustration of a Surjective but Not Injective Function
Definition of the Function:
Let f: R → R be defined by f(x) x2 - 1.
Explanation:
This function is surjective because it can produce every real number greater than or equal to -1. It is not injective because both f(1) 1 - 1 0 and f(-1) (-1)2 - 1 0. This demonstrates that the function is not one-to-one.
In conclusion, understanding the properties of surjective and injective functions, as demonstrated in these examples, provides a foundational framework for more advanced mathematical theories and applications.