Indexed Families of Sets

Set Theory · Axiom Academy

LESSON Indexed Families of Sets Working with collections of sets systematically When working with many sets, we need a systematic way to organize them. An indexed family assigns a "label" to each set using an index set . Suppose we have intervals for each positive integer n: We want to talk about the union of all these sets. But there are infinitely many! Use an index set I to organize these sets. We write to mean "the family of sets where i ranges over I". An indexed family of sets is a collection where: I is the index set (the set of labels) — natural numbers (countably infinite family) — real numbers (uncountably infinite family) The union of an indexed family is: An element x is in the union if it belongs to at least one . Every positive real number is in some interval (0, n) for large enough n. You may see the union written as: The intersection of an indexed family is: An element x is in the intersection if it belongs to every . Only 0 is in every interval (-1/n, 1/n) as they shrink toward 0! If the index set I is empty , the intersection is often defined as the universal set (everything satisfies "for all i ∈ ∅"). This is a convention that varies by textbook. De Morgan's Laws extend to indexed families: For an indexed family, we can form the Cartesian product : This is the set of all "choice functions" that pick one element from each . Indexed families let us work with arbitrarily many sets De Morgan's Laws generalize: complement swaps ∪ and ∩

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