Functions

Introduction and Definition of a Function

In mathematics, a function is a fundamental concept that describes a specific type of relation between elements of two sets, typically called the domain and the codomain. A function associates each element in the domain with a unique element in the codomain, and it is often represented as a set of ordered pairs.

In mathematics, a function $f$ from a domain set $A$ to a codomain set $B$ is defined as follows:

1. Each element $a$ in the domain set $A$ is associated with exactly one element $b$in the codomain set $B: [ \forall a \in A, \exists! b \in B \text{ such that } (a, b) \in f ]$
   Here, ∃! denotes “there exists a unique.”
2. The set of ordered pairs $(a, b)$ such that $a$ is in the domain $A$ and $b$ is in the codomain $B$ defines the function $f: [ f = {(a, b) \,|\, a \in A, b \in B} ]$

In simpler terms, a function takes an input from the domain, processes it in some way, and produces a unique output in the codomain for each input. The key characteristics of a function are that it must assign exactly one value to each element in the domain, and it must do so uniquely.

One-to-One (Injective) Function

An injective function, also known as a one-to-one function, is a type of function in mathematics that has the property that each distinct element in the domain is mapped to a distinct element in the codomain. In other words, for an injective function, no two different elements in the domain are assigned the same element in the codomain.

A function $f$ from a domain set $A$ to a codomain set $B$ is injective (one-to-one) if, for all distinct elements $a_1 and a_2$ in the domain $A$, their images under the function $f(a_1)$ and $f(a_2)$ in the codomain $B$ are also distinct:

$[ f(a_1) \neq f(a_2) \text{ whenever } a_1 \neq a_2 \text{ in } A ]$

In simpler terms, if you have two different inputs in the domain, they must produce two different outputs in the codomain. There is no “collapsing” or mapping of multiple domain elements to the same codomain element.

Injective functions are often depicted graphically as functions where no two distinct points in the domain map to the same point in the codomain. They are important in various areas of mathematics and have applications in many fields, including computer science, cryptography, and data analysis.

Onto (Surjective) Function

A surjective function, also known as an onto function, is a type of function in mathematics where every element in the codomain is mapped to by at least one element in the domain. In other words, a function is surjective if it covers the entire codomain, leaving no element without a pre-image in the domain.

A function $f$ from a domain set $A$ to a codomain set $B$ is surjective (onto) if, for every element $b$ in the codomain $B$, there exists at least one element $a$ in the domain $A$ such that $f(a) = b:$

$\forall b \in B, \exists a \in A \text{ such that } f(a) = b$

In simpler terms, a surjective function ensures that there are no “leftover” elements in the codomain without a corresponding element in the domain. Every element in the codomain is the result of applying the function to at least one element in the domain.

Surjective functions are often depicted graphically as functions where the entire codomain is covered or “hit” by the function. They are important in various mathematical and practical contexts, including function composition and solving equations.