Sets, Indexed Sets, Product Sets

Indexed Sets, Product Sets

A family of indexed sets has a set Ax associated with each x in some other set S. If S is the positive integers, then the family of sets includes A1, A2, A3, and so on.

The union of the family is the set of all elements y in Ax for any x in S. The intersection of the family is the set of all elements y in Ax for every x in S. Again, axioms insure the union and intersection of indexed sets are sets.

The product of indexed sets Ax for x in S is the set of functions on S that take each x into some member of Ax. For S finite, this is equivalent to the cross product of the individual sets. When all sets Ax are the same set A, we are taking A to the S power, written AS. If A is (0,1), AS represents the power set of S, sometimes written 2S. Collect members of S with f(x) = 1 to produce the subset associated with each function.

We're getting ahead of ourselves here, because set theoretic axioms have not yet been defined, but since we're talking about product sets, here it is.

The axiom of choice states that the product of indexed sets Ax across S is nonempty when S is nonempty and each Ax is nonempty. This axiom is consistent with the rest of set theory, and is usually assumed, because we like the theorems that result. They are often generalizations of theorems that are true for countable sets, or structures of countable dimension, etc. So why not bring in choice, and make the theorems true for all sets?

A finite product is nonempty; we can prove that from first principles. There is some y1 in A1, there is some y2 in A2, … there is some yn in An, and the tuple y1,y2,…yn is in the product set, hence the product is nonempty. This proof falls apart when the product is infinite, because the proof would require an infinite number of steps. The tuple is built element by element, and an infinite tuple demands an infinite chain of logical assertions. This is not a proof, hence we cannot prove that an infinite product is nonempty, at least not via this mechanism.

If the intersection of the family of indexed sets is nonempty, then so is the product. Let y be a member of every Ax. Now the function that maps each x in S to y is a well defined set, and is included in the product set, hence the product is nonempty.