Indexed Families of Sets

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(Indexed) Families of Sets

Section 5.5.

Examples

A Finite Family

Suppose A = { A₁, A₂, B }, where A₁ = {1,2,3}, A₂ = {1,3,5}, B = {2,4,6}.

Give an example of an element of A: {1,2,3} ∈ A

Note that the elements of A are sets, not the individual numbers or symbols we’ve seen before. This is what makes A a “family,” i.e., a set of sets.

What is A₁ ∪ A₂ ∪ B? A₁ ∪ A₂ ∪ B = {1,2,3,4,5,6}. Again because elements of a family of sets are sets, we can talk about doing set operations on them.

Finally, notice that while A is a family of sets, and even though two of its members’ names involve subscripts, it isn’t really an “indexed” family — there’s no indexing set, the subscripts aren’t in any way tied to the definitions of A₁ and A₂, etc. Not all families of sets have to be indexed.

An Infinite Family

Indexing often makes it easier to define and talk about a family though. For example...

For each natural number, n, let A_n = { x ∈ ℤ | x ≤ -n or x ≥ n }

What are A₁, A₂?

A₁ = { x ∈ ℤ | x ≤ -1 or x ≥ 1 } = { ..., -3, -2, -1, 1, 2, 3, ... }

A₂ = { x ∈ ℤ | x ≤ -2 or x ≥ 2 } = { ...,-4, -3, -2, 2, 3, 4, ... }

Define A = { A_i | i ∈ ℕ }

What is the indexing set, Λ, for this family?

Λ = ℕ

Note, however, that indexing sets don’t have to be sets of numbers, any non-empty set will do.

What is ∪_{n ∈ ℕ} A_n? ∪_{n ∈ ℕ} A_n = { ..., -3, -2, -1, 1, 2, 3, ... } since A_n+1 ⊂ A_n.

What is ∩_{n ∈ ℕ} A_n? ∩_{n ∈ ℕ} A_n = ∅ since for all natural numbers k, k not in A_k+1.

Notice that there is some reasoning about the definitions of the various elements of A behind these answers. While that reasoning isn’t necessarily a formal proof (although it could be), it does point out the desirability of writing math containing expressions related to indexed families. Here’s a brief example written with LaTeX, and its LaTeX source.

Key Ideas

Definitions, notations, especially “family”, “indexing set”, “index”

Reasoning about infinite indexed families via definitions of members in terms indices

Disjointness, and the idea that any pair of members of a family being disjoint is often more useful than whether the family as a whole is disjoint.

Problem Set

See handout for details.

Next

Functions as formal mathematical objects.

Read sections 6.1 and 6.2.

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