Relations

Functional [Function](https://en.wikipedia.org/wiki/Function_(mathematics)) Left-total Injective Surjective Bijection Transitive Reflexive Coreflexive Irreflexive Symmetric Antisymmetric Asymmetric Total Connex Idempotent Equivalence Preorder Partial order Total order Strict partial order Strict total order Dense Triadic relation

Binary relation

Algebraic logic [Allegory (category theory)](https://en.wikipedia.org/wiki/Allegory_(category_theory)) Cartesian product Cartesian square Cylindric algebras [Extension in logic](https://en.wikipedia.org/wiki/Extension_(predicate_logic)) [Involution](https://en.wikipedia.org/wiki/Involution_(mathematics)) Logic of relatives Logical matrix Predicate functor logic Quantale [Relation](https://en.wikipedia.org/wiki/Relation_(mathematics)) Relation construction Relational calculus Relational algebra Residuated Boolean algebra Spatial-temporal reasoning Theory of relations

$xRy : x \in X, y\in Y\\$ $R \subseteq X \times Y$ $X$ and $Y$ $X^2$ $xRx$

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.

Relations are categorized by the special properties they hold.

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 $x\in X$.

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: the Cartesian product itself forms a universal (full) relation and the empty set (being a subset of the Cartesian product set) forms an empty (or null) relation.

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

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.