Relations

The notion of relation is based upon the notion of Cartesian product, which in turn, is based on the notion of the ordered pairs.

The Cartesian product (dot product, the product) of two sets $A$ and $B$, denoted by $A \times B$ is the set of all ordered pairs $(a,b)$, with $a$ coming from the domain set $A$, and $b$ coming from the codomain set $B$.

Given two sets, $A$ and $B$, their dot product is denoted by:

$A \times B = \{ \forall a \in A \land \forall b \in B \mid (a,b) \}$

Informally, the Cartesian product between two sets is the matching of each element in the domain with each element in the codomain. The elements are matched as part of the new structure called an ordered pair.

A={x,y} ∧ B={1,2} -> A x B = { (x,1), (x,2), (y,1), (y,2) }

In such an ordered pair, the pair of objects is denoted $(a,b)$, where the object $a$ is called the first component, and the object $b$ is the second component of the ordered pair. An ordered pair is an object in its own right. An ordered pair is the constituent member of the Cartesian product, and with that of each and every relation.

Relation

Any relation between two sets is a subset of their Cartesian product:

$x \in X, y\in Y \mid xRy \mid R \subseteq X \times Y$

Unlike an unordered pair, which is just a plain set consisting of a couple of elements, $\{a,b\}$, and hence does not have any notin of order, an ordered pair is affected by the order of its two components, so that

$\{a,b\} \neq (a,b)\neq (b,a)$

A binary relation $R$ from set $X$ to set $Y$, where $x \in X$ and $y \in Y$, is denoted as $xRy$, and it is a subset of the Cartesian product $X \times Y$.

so, a binary relation $R$ on the sets $A$ and $B$ is in fact an element in their power set:

$R \in \mathfrak{P}(A \times B)$

The sets $X$ and $Y$ may be the same set $X$ in which case its Cartesian product is usually denoted by $X^2$ and relations by $xRx$.

Binary relations are used to model concepts like "is greater than", "is equal to", and similar. The concept of function is defined as a special kind of binary relation.

A binary relation between sets $A$ and $B$ is a subset of their Cartesian product, $A \times B$. Or equivalently, it is an element in the powerset of their Cartesian product.

Any subset of the Cartesian product forms a relation, whether it is a named (e.g. symmetric) or unnamed relation; even the empty set (being a subset of the Cartesian product set) is a relation.

A single set's Cartesian product (with itself) is commonly denoted as $\mathbb{N^2} = \mathbb{N} \times \mathbb{N}$.

If $X$ and $Y$ are sets, the Cartesian product $X \times Y$ is the set of all ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. And the set $X^2 =X\times X$ is the set where all pair of $x\in X$.

The set $\mathbb{N^2} = \mathbb{N} \times \mathbb{N}$ of ordered pairs of natural numbers (starting and ending curly-braces demarking this set are not showndue to formatting):

$$\begin{matrix} (1,1), &(1,2), &(1,3), &\dots \\ (2,1), &(2,2), &(2,3), &\dots \\ (3,1), &(3,2), &(3,3), &\dots \\ \vdots &\vdots &\vdots &\ddots \end{matrix}\\ \text{figure 1.1}$$

Any subset of this set forms a relation:

• full relation, where every pair participates, is the set of the Cartesian product itself.

• on the other side of the extreme is the empty relation which is the empty set; even though no pair participates, it is still considered a relation.

• between these two extremes are all other relations, some of which have a name, being more popular then others. The most popular ones, come with a name and a special symbol attached.

Less than (LT, <) relation is formed by the subset of all pairs lying above the diagonal, and greater than (GT, >) by the subset of all pairs below the diagonal. The union of these two with identity relation form less than or equal to (LE, <=) and greater than or equal to (GE, >=) relations, respectively.

Relations are categorized by the special properties they hold. Some are very important like orders and equivalence relations.

Types of Relations

(using figure 1.1 as example)

• Full relation is the Cartesian product set itself, $R=\mathbb{N^2}$

• Empty relation is the empty set, $R=\emptyset$

• Identity relation on $\mathbb{N}$: the subset consisting of the pairs lying on the diagonal i.e. $\{(1,1),(2,2),(3,3),\dots\}$. It can be defined as $Id_x=\{(x,x) : x \in X\}$

• Inverse relation is the set of inverted pairs, usually denoted by $R'$.

References

https://ncatlab.org/nlab/show/relation