# Relations

[Functional](https://en.wikipedia.org/wiki/Functional_relation) \[Function]\(<https://en.wikipedia.org/wiki/Function_(mathematics>)) [Left-total](https://en.wikipedia.org/wiki/Binary_relation) [Injective](https://en.wikipedia.org/wiki/Injective) [Surjective](https://en.wikipedia.org/wiki/Surjective) [Bijection](https://en.wikipedia.org/wiki/Bijection) [Transitive](https://en.wikipedia.org/wiki/Transitive_relation) [Reflexive](https://en.wikipedia.org/wiki/Reflexive_relation) [Coreflexive](https://en.wikipedia.org/wiki/Coreflexive_relation) [Irreflexive](https://en.wikipedia.org/wiki/Irreflexive_relation) [Symmetric](https://en.wikipedia.org/wiki/Symmetric_relation) [Antisymmetric](https://en.wikipedia.org/wiki/Antisymmetric_relation) [Asymmetric](https://en.wikipedia.org/wiki/Asymmetric_relation) [Total](https://en.wikipedia.org/wiki/Total_relation) [Connex](https://en.wikipedia.org/wiki/Total_relation) [Idempotent](https://en.wikipedia.org/wiki/Idempotent_relation) [Equivalence](https://en.wikipedia.org/wiki/Equivalence_relation) [Preorder](https://en.wikipedia.org/wiki/Preorder) [Partial order](https://en.wikipedia.org/wiki/Partial_order) [Total order](https://en.wikipedia.org/wiki/Total_order) [Strict partial order](https://en.wikipedia.org/wiki/Strict_partial_order) [Strict total order](https://en.wikipedia.org/wiki/Total_order) [Dense](https://en.wikipedia.org/wiki/Dense_order) [Triadic relation](https://en.wikipedia.org/wiki/Triadic_relation)

[Binary relation](https://en.wikipedia.org/wiki/Binary_relation)

[Algebraic logic](https://en.wikipedia.org/wiki/Algebraic_logic) \[Allegory (category theory)]\(<https://en.wikipedia.org/wiki/Allegory_(category_theory>)) [Cartesian product](https://en.wikipedia.org/wiki/Cartesian_product) [Cartesian square](https://en.wikipedia.org/wiki/Cartesian_square) [Cylindric algebras](https://en.wikipedia.org/wiki/Cylindric_algebra) \[Extension in logic]\(<https://en.wikipedia.org/wiki/Extension_(predicate_logic>)) \[Involution]\(<https://en.wikipedia.org/wiki/Involution_(mathematics>)) [Logic of relatives](https://en.wikipedia.org/wiki/Logic_of_relatives) [Logical matrix](https://en.wikipedia.org/wiki/Logical_matrix) [Predicate functor logic](https://en.wikipedia.org/wiki/Predicate_functor_logic) [Quantale](https://en.wikipedia.org/wiki/Quantale) \[Relation]\(<https://en.wikipedia.org/wiki/Relation_(mathematics>)) [Relation construction](https://en.wikipedia.org/wiki/Relation_construction) [Relational calculus](https://en.wikipedia.org/wiki/Relational_calculus) [Relational algebra](https://en.wikipedia.org/wiki/Relational_algebra) [Residuated Boolean algebra](https://en.wikipedia.org/wiki/Residuated_Boolean_algebra) [Spatial-temporal reasoning](https://en.wikipedia.org/wiki/Spatial-temporal_reasoning) [Theory of relations](https://en.wikipedia.org/wiki/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.
