Questions: Indexed Families and Generalized Operations

Question 1

Let S_n = {0, 1, 2, ..., n} for each n ∈ ℕ. What is ⋃_{n∈ℕ} S_n?

• A. The empty set ∅, because no single element belongs to every S_n
• B. ℕ itself, because every natural number k belongs to S_k and hence to at least one set in the family
• C. The set ℕ, but only because all the S_n are distinct sets
• D. An undefined 'S_∞' — the infinite limit of the family

Answer: (B)

Generalized union ⋃_{i∈I} S_i contains every element belonging to at least one S_i. Every natural number k belongs to S_k (since S_k = {0, 1, ..., k} contains k), so k is in the union. Therefore ⋃_{n∈ℕ} S_n = ℕ. Option A is the classic confusion between union and intersection — it is the intersection that requires membership in every set. Option C's qualifier 'only because they are distinct' is irrelevant; union is well-defined regardless. Option D introduces a nonexistent 'S_∞'; no such set is in the indexed family.

Question 2

Let S_n = (0, 1/n] for each n ∈ ℕ⁺. What is ⋂_{n∈ℕ⁺} S_n?

• A. (0, 1] — the first and largest interval in the family
• B. The empty set ∅ — no positive real number belongs to every S_n
• C. {0} — zero is the only value common to all intervals
• D. A degenerate interval (0, 0] containing only values infinitely close to 0

Answer: (B)

For any positive real number x > 0, choose n large enough that 1/n < x (always possible by the Archimedean property). Then x ∉ S_n = (0, 1/n], so x cannot be in the intersection. Since this holds for every x > 0, and 0 ∉ S_n for any n (the intervals are open at 0), no real number belongs to every S_n. The intersection is ∅. This is a classic example showing that an infinite intersection of nonempty sets can be empty — a result that only becomes rigorous through the indexed family framework, which allows the universal quantifier 'for all i ∈ I' to range over an infinite set.

Question 3

In an indexed family {S_i : i ∈ I}, two distinct indices i ≠ j are allowed to refer to the same set (S_i = S_j).

• T. True
• F. False

Answer: True

An indexed family is formally a function f: I → V, and a function can assign the same value to multiple inputs. For example, the family {S_i : i ∈ ℤ} where S_i = {0} for all i is a perfectly valid indexed family even though every member is identical. The indexing provides a mechanism for labeling and counting sets in a collection; it does not require the sets to be distinct. This flexibility matters when indexing over complex index sets where multiple indices naturally map to the same set.

Question 4

A generalized intersection ⋂_{i∈I} S_i is mainly well-defined when the index set I is finite.

• T. True
• F. False

Answer: False

The generalized intersection is defined for any index set I, including countably infinite and uncountably infinite ones. The definition x ∈ ⋂_{i∈I} S_i if and only if for all i ∈ I, x ∈ S_i applies regardless of the cardinality of I. The intersection of an infinite family is equally well-defined as the finite case — it is just a universal quantifier ranging over all i ∈ I. The example S_n = {n, n+1, ...} with intersection ∅ demonstrates a well-defined infinite intersection with an informative result.

Question 5

Explain why an indexed family is formally defined as a function, and what this formalism adds over simply saying 'a collection of sets.'

Model answer: Defining an indexed family as a function f: I → V gives three advantages over an informal collection. First, it allows the same set to appear multiple times under different indices (since functions can map multiple inputs to the same output), which an unordered collection cannot represent. Second, it lets the family inherit the structure of the index set I — for example, if I = ℕ, the family is ordered and the notion of 'the nth set' is precise. Third, it enables rigorous generalization to arbitrary cardinalities: a function f: I → V is well-defined whether I is finite, countably infinite, or uncountably infinite, whereas 'a collection' has no built-in mechanism for infinite or uncountable cases.

The deeper reason is foundational: 'a collection of sets' is informal and not a set-theoretic object. A function f: I → V is a precise object whose existence can be verified in set theory using the axioms. When sequences, Cartesian products, and topological constructions are all defined as indexed families, they inherit this rigor. The formalism also makes it clear what operations are allowed: you can compose functions, restrict them to subsets of I, and use them in formal proofs — none of which is straightforward with an informal 'collection.'