Definition:Cartesian Product of Relations

Definition

Let $\family {S_i}_{i \mathop \in I}$ and $\family {T_i}_{i \mathop \in I}$ be families of sets indexed by $I$.

For each $i \in I$, let $\RR_i \subseteq S_i \times T_i$ be a relation from $S_i$ to $T_i$.

Let $S$ and $T$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$ and $\family {T_i}_{i \mathop \in I}$ respectively:

$\ds S = \prod_{i \mathop \in I} S_i$

$\ds T = \prod_{i \mathop \in I} T_i$

Then the product of the relations $\RR_i$ is defined as the relation $\RR \subseteq S \times T$ such that:

$x \mathrel \RR y \iff \forall i \in I: x_i \mathrel {\RR_i} y_i$

Also see

• Product of Transitive Relations is Transitive
• Product of Reflexive Relations is Reflexive
• Product of Antisymmetric Relations is Antisymmetric
• Product of Preorders is Preorder
• Product of Orders is Order
• Product of Directed Sets is Directed Set