Functions and sets

A function can be regarded as a mapping between sets. Functions are a central concept in mathematics because they describe how each element in one set corresponds to exactly one element in another set.

Functions as mappings

A function $ \large f$ from a set $ \large A$ to a set $ \large B$ is a rule that assigns to each element in $ \large A$ exactly one element in $ \large B$. It is written as:

$$ \large f : A \to B $$

If $ \large a \in A$, then the image of $ \large a$ under $ \large f$ is written as $ \large f(a)$.

Domain and codomain

For a function $ \large f : A \to B$:

$ \large A$ is called the domain.
$ \large B$ is called the codomain.
The set of all actual function values $ \large f(a)$ is called the range (or image).

Example: If $ \large f(x) = x^2$ with domain $ \large \mathbb{R}$, then the range is $ \large [0,\infty [$.

Applications

Functions play a central role in both mathematics and computer science.

In mathematics, they are used to describe relationships between variables, to set up models, and to perform calculations.
In computer science, functions serve as building blocks in programming languages, algorithms, and databases. Many computations can be seen as functions translating input into output.
In cryptography and information security, functions with special properties (e.g. bijective functions and hash functions) are fundamental.

Understanding functions is therefore necessary both to master mathematics and to work with technology.