Discrete Math: Relations

Question

Why is it necessary for a relation to be a subset of the Cartesian product of two sets. Why couldn't we say that a relation is a relationship between any two elements of one or more sets.

Answer 1

I'm going to answer this on the presumption that it's sort of a (soft-question).

Relations are defined to be a subset of the Cartesian product of two sets, which may not be distinct, because such a definition is simple and yet fully expressive. In other words, by defining relations that way, we can get everything else that you describe.

For instance, if we call our sets $A_1, A_2, A_3, \ldots$, then we can let $A = A_1 \cup A_2 \cup A_3 \cup \cdots$, and then a relation as you would like to define it would be a subset of $A \times A$.

It's analogous to the way that (say) integer addition is defined first as a binary operation: We define it as a function $+:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$. We define it this way despite the fact that we all have seen and know how to deal with expressions such as $1+2+3+4$: because the definition of binary addition, in conjunction with associativity and additive identity, allows us to add one or more integers; and because defining it this way allows us to use all the other things we've learned about binary operations and prove things about it more easily.

Answer 2

A relation is defined as a subset of the Cartesian product of some fixed number of sets. The word "necessary" in your question is thus irrelevant, as it could've also been defined in other ways.

If, for example, the "relationship" you're referring to is a binary one (i.e. two elements can either be in a relationship together, or not), then defining a relation as a subset of a Cartesian product is probably the most "natural" way to do so.

If this "relationship" is more involved, then perhaps you are looking for something else.