# 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 $$(x,y)\in R \land (y,z)\in S$$. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/220-relation-theory/terms/composition-of-relations.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.