## Relation concatenation

### Set

context | $ R \in \text{Rel}(X,U) $ |

context | $ S \in \text{Rel}(V,Y) $ |

definiendum | $ \langle x,y \rangle \in S\circ R $ |

postulate | $ \exists m.\ \langle x,m \rangle \in R \land \langle m,y \rangle \in S $ |

### Discussion

Concatenations/compositions are associative.

A mayority of uses of the relation concatenation is when the relation is functional, i.e. one composes functions alla

$g:X\to Y,\ \ f:Y\to Z$

then

$f\circ g:X\to Z$

$(f\circ g)(x):=f(g(x))$

#### Notation

If $f:X\to Y$ and $g:Y\to Z$ are functions, we'll often denote $f\circ g$ by $fg$. This convenient notation will also be used in more elaborate cases. For example, if by $f(x)$ we done the values of a function $f:X\to\mathbb R$ and $|\cdot|$ is the function which takes a real to its absolute value, then $|f|$ will denote the name of the function with values $|f(x)|$.