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Home , Family of sets, Indexed family

of Sets, Indexed Families of Sets

Philippe B. Laval

KSU

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Philippe B. Laval (KSU) Sequences of Sets Today 1 / 14 Introduction

We extend the definition of sequences studied in Calculus to sequences of sets and indexed families of sets. We then prove that the main properties of the operations on two sets extend to sequences of sets.

Philippe B. Laval (KSU) Sequences of Sets Today 2 / 14 Sequences: Quick review

Definition () Let S R. ⊆ 1 A sequence in S is a f : K S where → K = n N : n n0 for some n0 N . { ∈ ≥ ∈ } th 2 For each n K, we let xn = f (n). xn is called the n term of the sequence f .∈

For convenience, we usually denote a sequence by xn ∞ rather than f . { }n=n0 Some texts also use (x )∞ . The starting point, n is usually 1 in which n n=n0 0 case, we simply write xn or (xn) to denote a sequence. In theory, the starting point does not{ have} to be 1. However, it is understood that whatever the starting point is, the elements xn should be defined for any n n0. ≥

Philippe B. Laval (KSU) Sequences of Sets Today 3 / 14 Sequences: Quick review

A sequence can be given different ways.

1 List the elements. 2 Give a formula to generate the terms. 3 A sequence can be given recursively. Like a function, a sequence can be plotted. However, since the domain is a of Z, the plot will consist of dots instead of a continuous curve. We now focus on sequences of sets and indexed families of sets.

Philippe B. Laval (KSU) Sequences of Sets Today 4 / 14 Sequences of Sets

Sequences of sets are similar to sequences of real numbers but the terms of the sequence are sets. The notation used is also similar. Definition Let X R, X = ∅ and let K = n N : n n0 for some n0 N .A sequence⊆ of subsets6 of X is a function{ ∈ f : K≥ P (X ), the power∈ } of X . → For each n K, we let An = f (n) be a subset of X that is an of P (x). ∈

Example

Consider Nn ∞ where Nn = 1, 2, 3, ..., n . Nn ∞ is a sequence of { }n=1 { } { }n=1 of N. Example 1 1 For each n N, define In = x R : 0 < x < = 0, . In ∞ is a ∈ ∈ n n { }n=1     sequence of subsets of R. Philippe B. Laval (KSU) Sequences of Sets Today 5 / 14 Indexed Families of Sets

The index we use for the subsets does not have to be an . It can be an element of any set. In this case, instead of calling it a sequence of sets, we call it an indexed . We give a precise definition. Definition Let A and X be non-empty sets. An of subsets of X with index set A is a function f : A P (X ). Like for sequences, if → f : A P (X ), then for each α A, we let Eα = f (α). We use a notation → ∈ similar to sequences that is we denote the indexed family Eα α A. { } ∈ Remark A sequence of sets is just an indexed family of sets with N as index set.

Example

For each x (0, 1), define Ex = r Q : 0 r < x . Then, Ex x (0,1) is ∈ { ∈ ≤ } { } ∈ an indexed family of subsets of Q. The index set is (0, 1).

Philippe B. Laval (KSU) Sequences of Sets Today 6 / 14 and Intersection of a Sequence of Subsets

The remainder of this section deals with sequences of sets, though the results and definitions given can be extended to indexed families of subsets. Definition (Union of a Sequence of Subsets)

Let An be a sequence of subsets of a set X . We define { } n Ai = A1 A2 ... An ∪ ∪ ∪ i=1 [ = x X : x Ai for some 1 i n { ∈ ∈ ≤ ≤ } Similarly, we define the union of the entire sequence by

∞ Ai = x X : x Ai for some natural number i { ∈ ∈ } i=1 [

Philippe B. Laval (KSU) Sequences of Sets Today 7 / 14 Union and Intersection of a Sequence of Subsets

Definition (Intersection of a Sequence of Subsets)

Let An be a sequence of subsets of a set X . We define { } n Ai = A1 A2 ... An ∩ ∩ ∩ i=1 \ = x X : x Ai for every natural number 1 i n { ∈ ∈ ≤ ≤ } and ∞ Ai = x X : x Ai for every natural number i { ∈ ∈ } i=1 \

Philippe B. Laval (KSU) Sequences of Sets Today 8 / 14 Union and Intersection of a Sequence of Subsets

Example

∞ ∞ Consider Nn ∞ where Nn = 1, 2, 3, ..., n . Find Ni and Ni . { }n=1 { } i=1 i=1 [ \ Example

1 ∞ For each n N, define In = x R : 0 < x < . Prove that Ii = ∅ ∈ ∈ n i=1   \ ∞ and Ii = I1. i=1 [

Philippe B. Laval (KSU) Sequences of Sets Today 9 / 14 Theorems

Theorem (Distributive Laws)

Let En and E be subsets of a set X . Then,

∞ ∞ 1 E Ei = (E Ei ) ∩ ∩ i=1 ! i=1 [ [ ∞ ∞ 2 E Ei = (E Ei ) ∪ ∪ i=1 ! i=1 \ \

Philippe B. Laval (KSU) Sequences of Sets Today 10 / 14 Theorems

Theorem (De Morgan’sLaws)

Let En be a sequence of subsets of X . Then, { } c ∞ ∞ 1 c Ei = Ei i=1 ! i=1 [ c \ ∞ ∞ 2 c Ei = Ei i=1 ! i=1 \ [

Philippe B. Laval (KSU) Sequences of Sets Today 11 / 14 Theorems

Theorem Let f : X Y → 1 If En is a sequence of subsets of X , then { }

∞ ∞ ∞ ∞ f Ei = f (Ei ) and f Ei f (Ei ) ⊆ i=1 ! i=1 i=1 ! i=1 [ [ \ \ 2 If Gn is a sequence of subsets of Y , then { }

1 ∞ ∞ 1 1 ∞ ∞ 1 f − Gi = f − (Gi ) and f − Gi = f − (Gi ) i=1 ! i=1 i=1 ! i=1 [ [ \ \

Philippe B. Laval (KSU) Sequences of Sets Today 12 / 14 Contracting and Expanding Sequences

Definition (Contracting and Expanding Sequences of Sets)

Suppose that An is a sequence of sets. { } 1 We say that An is expanding if we have An An+1 for every n in the domain of{ the} sequence. ⊆

2 We say that An is contracting if we have An+1 An for every n in the domain of{ the} sequence. ⊆

Example 1 1 Is the sequence of sets Jn where Jn = , . Jn contracting or { } −n n expanding?  

Example

Is the sequence of sets Kn where Kn = ( n, n). Kn contracting or expanding? { } −

Philippe B. Laval (KSU) Sequences of Sets Today 13 / 14 Exercises

See the problems at the end of my notes on sequences of sets.

Philippe B. Laval (KSU) Sequences of Sets Today 14 / 14