binary relation

A binary relation relates some elements of one set called the domain with some elements of another set (possibly the same) called the codomain.

Formally,
a binary relation $R$ from domain $X$ to codomain $Y$ is a set of ordered pairs

R = {(x, y) \in X \times Y : x R y}

where $X \times Y$ is the Cartesian product of $X$ and $Y$ ,
and $x R y$ is infix notation for " $x$ is $R$ -related to $y$ ".

The domain of definition of $R$ is the set of all $x$ such that $x R y$ for at least one $y$ .
The codomain of definition of $R$ is the set of all $y$ such that $x R y$ for at least one $x$ .

A binary relation can be:

injective relation (left-unique)
functional relation (right-unique)
total relation (left-total)
surjective relation (right-total)

We distinguish homogeneous and heterogeneous relations by whether their domain and codomain are the same or distinct, respectively.

Calculus of relations includes set operations, composition of relations, converse relation

A function is a binary relation that is functional and total.

A bijection is a function which pairs every element to exactly one other element.