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> [!definition] Definition. ([[relation]])
> A **relation** on a is a subset $C$ of a [[cartesian product]] $A \times B$. (Often $B=A$ and we say $C$ "is a relation on $A
quot;.)
\
For a **relation** $C \subset A \times B$ we use the notation $xCy$ to indicate $(x,y)\in C$, read "$x$ is in the relation $C$ to $yquot;.
>
The [[category]] of sets and relations is denoted $\mathsf{Rel}$. Its objects are sets $X$, morphisms $R:X \to Y$ are relations $R \subset X \times Y$, and composition $X \xrightarrow{R}Y \xrightarrow{S}Z$ is given by $S \circ R=\{ (x, z) : xRy \text{ and }ySz \, \, \, \,(\ex y \in Y) \} \subset X \times Z.$
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####
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```