Function

$X\times Y = \{\forall xy \to x\in X\land y\in Y\ \to (x,y=f(x))\}$

$\forall x \in X \to \exists y \in Y \to (x,y) \in f$

$\forall x \in X \to \exists y \in Y \to (x,y) \in f$

$\forall x \in X \to \exists y \in Y \to f(x)=y$

A function is a relation (process) that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function.

If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).

A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function.

When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane.

In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere.

Links and References

https://en.wikipedia.org/wiki/Function_(mathematics)