Surjective function

Surjective function, also known as an onto function, is a fundamental concept in mathematics, particularly in the field of functions, set theory, and calculus. A function \(f\) from a set \(X\) to a set \(Y\) is called surjective if for every element \(y\) in the set \(Y\), there is at least one element \(x\) in the set \(X\) such that \(f(x) = y\). In simpler terms, a surjective function covers the entire target set \(Y\).

Definition

Formally, a function \(f : X \rightarrow Y\) is surjective if for every \(y \in Y\), there exists at least one \(x \in X\) such that \(f(x) = y\). This concept is one of the key types of functions studied in mathematics, the others being injective (one-to-one) functions and bijective (both injective and surjective) functions.

Examples

1. The function \(f(x) = x^2\) from \(\mathbb{R}\) (the set of all real numbers) to \(\mathbb{R}^+\) (the set of all positive real numbers including zero) is surjective because every positive number or zero has a real square root. 2. The function \(f(x) = 2x + 3\) from \(\mathbb{R}\) to \(\mathbb{R}\) is surjective because, for every real number \(y\), there is a real number \(x\) such that \(2x + 3 = y\).

Properties

- A function is surjective if its codomain equals its range. - Surjectivity is a property that depends on both the function and its codomain. Changing the codomain can change whether a function is surjective. - In compositions of functions, if \(g \circ f\) is surjective, then \(g\) is surjective, but \(f\) need not be. - The inverse of a surjective function, if it exists, is not necessarily a function but a relation.

Surjectivity in Other Areas

Surjective functions are not only studied in pure mathematics but also have applications in other areas such as computer science, where they are related to concepts like hashing and data structure design, and in economics, where functions representing supply and demand curves can be considered surjective under certain conditions.

See Also

• Function
• Injective function
• Bijective function
• Function composition
• Inverse function