Composition of relations

https://en.wikipedia.org/wiki/Composition_of_relations

In binary relations, the composition of relations refers to the formation of a new relation, denoted by R ; S, given two relations R and S.

The composition of relations is called relative multiplication in the calculus of relations. The composition is then the relative product of the factor relations.

Composition of functions is a special case of composition of relations.

The words "uncle" indicates a compound relation: for a person to be somebody's uncle, he first must be a brother of a parent. In algebraic logic it is said that the relation of "is an uncle of", (xUz), is the composition of relations "is a brother of", xBy, and "is a parent of", yPz.

$xBy \land yPz \iff xUz$

x is an uncle of z if and only if x is a brother of y who is a parent of z

Definition

If $R\subseteq X\times Y \land S\subseteq Y\times Z$ are two binary relations, then their composition, $R; S$ is the relation:

$$\displaystyle R;S = \{(x,z) \in X \times Z \ . \ \exists y \in Y \ .\ (x,y) \in R \land (y,z) \in S \}$$

In other words, $R;S\subseteq X\times Z$ is defined by the rule that says $(x,z)\in R;S$ iff there is an element $y\in Y$ such that $x\,R\,y\,S\,z$, that is $(x,y)\in R \land (y,z)\in S$.