Subsection 7.3.10 (00QC): Products of Families of Relations

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Style

Class

Font

Theorem Environments

Style 1

book

Alegreya Sans

tcbthm

Style 2

book

Alegreya Sans

amsthm

Style 3

book

Arno*

amsthm

Style 4

book

Computer Modern

amsthm

Let

{A_{i}}_{i \in I}

and

{B_{i}}_{i \in I}

be families of sets, and let

{R_{i} : A_{i} \to | B_{i}}_{i \in I}

be a family of relations.

The product of the family ${R_{i}}_{i \in I}$ is the relation $\prod_{i \in I} R_{i}$ from $\prod_{i \in I} A_{i}$ to $\prod_{i \in I} B_{i}$ defined as follows:

Viewing relations as subsets, we define $\prod_{i \in I} R_{i}$ as its product as a family of sets, i.e. we have

$\prod_{i \in I} R_{i} \overset{def}{=} {(a_{i}, b_{i})_{i \in I} \in \prod_{i \in I} (A_{i} \times B_{i}) | \begin{aligned} for each i \in I, \\ we have a_{i} \sim_{R_{i}} b_{i} \end{aligned}} .$
Viewing relations as functions to powersets, we define

$[\prod_{i \in I} R_{i}] ((a_{i})_{i \in I}) \overset{def}{=} \prod_{i \in I} R_{i} (a_{i})$

for each $(a_{i})_{i \in I} \in \prod_{i \in I} R_{i}$ .

The Clowder Project

7.3.10 Products of Families of Relations