2.2
Families of Sets

2.2.1 Introduction

The union and intersection of sets can be extended to the union and intersection of many sets, even an infinite number. When dealing with a collection of several sets, it is usual practice to refer to them as families or classes of sets. To denote a family of sets, one often uses indices such as {A1, A2, A3, … , A10} and for an infinite family, we might write {A1, A2, A3, …} or

equation

Other common ways to denote families of sets are

equation

where the set Λ is called an index set.

For example

equation

The reader might recall the notation for infinite sums and products as

equation

which motivates the following notation and definition for sets.

The following examples illustrate these ideas.

Diagram of increasing family of closed intervals displaying a number line with brackets enclosing broad line from 0 to 1/2 for A2, 0 to 2/3 for A3, 0 to 3/4 for A4, and 0 to k-1/k for Ak from top to bottom.

Figure 2.8 Increasing family of closed intervals.

Plane of projection of a set displaying a broad y-axis and an ellipse labeled S enclosing vertical lines with arrows marking A1, A2, and A3.

Figure 2.10 Projection of a set.

2.2.2 Extended Laws for Sets

Many rules for the intersection and union of sets introduced in Section 2.1 can easily be extended to families of sets. We leave the proofs to many of these laws to the reader.

Proof of d)

(⊆): To show images we let

equation

Hence,

equation

(⊇): The proof of the set containment ⊇ as well as the proofs of a), b), and c) follows along similar lines and is left to the reader.

2.2.3 Topologies on a Set

A topology on a set is a family of subsets of the set that places a “structure” on the set that allows for the study of convergence and limits of points in the set. The study of point‐set topology, which we introduce in Section 5.4, forms the “metrical” structure for several areas of mathematics, such as real and complex analysis.

The idea is to introduce a family J of subsets of a given set U, such as a family J of subsets of the real numbers U = ℝ. The sets in the family J are called open sets, and these sets act as “neighborhoods” of points in U, allowing for the discussion of convergence sequences in U. The family of open sets J is called a topology on U. But not any collection of subsets of U is a topology. There are three restrictions on a family J in order that it be a topology on U, They are as follows:

Problems

  1. Unions and Intersections

    Let A1 = {1, 2}, A2 = {2, 3}, A3 = {3, 4} and in general Ak = {k, k + 1}. Write explicitly the following sets:

    1. images
    2. images
    3. images
    4. images
    5. images
    6. images
  2. More Unions and Intersections

    Find the infinite union and intersections

    equation

    of the following sets:

    1. images
    2. images
    3. images
    4. images
    5. Ak = [k, k + 1]
    6. images
  3. Families of Sets in the Plane

    Define a family of subsets

    equation

    of the plane ℝ2 by where m, n ∈ ℕ. Find the following sets. Hint: Proceed like one does with double series.

    1. images
    2. images
  4. Identity of an Indexed Family

    Prove the following distributive property for indexed families of sets:

    equation
  5. Algebra of Sets

    Let A be a set and a collection of subsets of A. The collection is called an algebra1 of sets if

    • c) C ∪ D is in whenever C and D are in
    • d) images is in whenever C is in

    When this happens, we say the family is closed under unions and complementation. Which of the following collections of subsets of A = {a, b, c} constitute an algebra of subsets of A ?

    1. The power set = P(A)
    2. = {Ø, A}
    3. = {Ø, {a}, A}
    4. = {Ø, {a}, {b, c}, A}
  6. Sets of Length Zero

    In measure theory, a subset A of the real numbers is said to have length (or measure) zero if ∀ε > 0 and there exists a sequence of intervals Ak = (ak, bk) that satisfy

    equation

    where their total length is less than ε; that is

    equation

    Show that any sequence of real numbers {ck, k = 1, 2, …} has measure zero. Hint: Cover {ck} by a union of intervals (ak, bk), where the length of each interval satisfies |bk − ak| = ε/2k. Use this result to show that the rational numbers have a total length of 0.

  7. Compact Sets

    A subset A of the real numbers is said to be compact if for every collection = {(aα, bα) : α ∈ Λ} of open intervals whose union contains (or covers) A; i.e.

    equation

    there exists a finite subcollection of intervals of whose union also contains (or covers) A. Show the set A = (0, 1) is not compact by showing the following.

    1. A is covered by

      images

    2. There does not exist a finite subcollection of whose union contains A.
  8. Topologies I

    Verify that the following families of subsets of A form topologies on A = {a, b, c}.

    equation
    equation
  9. Topologies II

    Which of the families of subsets of {a, b, c} are topologies on {a, b, c}?

  10. Finding Intersections

    Find an infinite family of sets whose intersection is

    1. {1}
    2. [0, ∞)
  11. Finding Unions

    Find an infinite family of sets whose union is

    1. (0, ∞)
  12. Internet Research

    There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward unions and intersections of families of sets, topology on a finite set.

Note

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