Indexed Families of Sets

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Questions?

Indexed Families of Sets

Section 5.5.

Idea

Describe an indexed family of sets containing 4 members whose i^th member is the set containing -i, 0, and i. What is the indexing set for your family?

Solution 1

Here is one solution, in which sets are indexed by the letters “b,” “c,” “d,” and “e,” which is fine although it makes the notion of the “i^th set” a bit hard to follow (ideally indices are values that make intuitive sense in whatever context the indexed family arises in).

A_b = {-1,0,1}

A_c = {-2,0,2}

A_d = {-3,0,3}

A_e = {-4,0,4}

{ A_a | a ∈ Λ }, where Λ = {b,c,d,e}

= { A_b, A_c, A_d, A_e } = { {-1,0,1}, {-2,0,2}, {-3,0,3}, {-4,0,4} }

Note that despite all the new terminology and notation, the indexed family of sets just comes down to a set of sets.

Solution 2

Here’s a solution with indices that capture the idea of “i^th set” a bit more closely.

A_f = {-5,0,5}, A₆ = {-6,0,6}, A₇ = (-7,0,7}, A_π = {-π,0,π}

Λ = { f, 6, 7, π }

This family is { A_f, A₆, A₇, A_π }, again really just a set of sets.

Operations

Consider the indexed family {A_i | i ∈ ℕ} with A_i = { 0, i² }. What is the union of all of the A_i? What about the intersection?

An example of a member of this family: {0,4}, which would be set A₂.

Since each member of the family contains 0² and i² for some natural number i, and every natural number is the index of some member, the union of all the members is the set of perfect squares:

Note that unions or intersections over an indexed family are unions or intersections of the members, and so produce a set of their elements, not another indexed family.

The intersection of all the members is the set {0}, since it’s the only element that appears in every member of the family.

Letting the universal set be the non-negative integers, is the same (by de Morgan’s Law) as the complement of the intersection, i.e., the set of all non-negative integers except 0.

Key Points

Indexed families as sets of sets with subscripts (aka the index) to pick out individual elements.

Operations on the members of indexed families, particularly union and intersection.

Next

Introduction to functions.

Read section 6.1.

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