In the previous post we talked about ordered pairs, Cartesian product. With it, we established the mathematical theory of relationships, using set theory. Motivation Initially, we consider the set $A \times B,$ in which $A = \\{\text{mulheres} \\}$ $B = \\{\text{homens} \\}$. In this case, we can make the sentence: Lucas is Rosana’s husband. Relationship of “being the husband of” Ordered pair $(a,b)$ $a = $ Lucas, $b = $ Rosana. Note that the pair $(b, a)$ does not belong to the relationship, as Rosana is not Lucas’s husband. If the relationship was “being married to”, then both couples would be in the relationship. Formalization Definition (Relation) A relationship between two sets $A$ and $B,$ denoted by $\mathcal{R}(A, B),$ is a subset of $A \times B.$