Download Section 2.2 Families of Sets

1
Section 2.2
Families of Sets
Section 2.2
2.2 Families of Sets
Purpose of Section: To introduce the concept of classes or families of sets
and show how operations on sets are applied to classes.
Introduction
Often in such areas of mathematics as point-set topology, measure
theory, and abstract algebra one often works with sets of sets, generally
called families of sets. Normally, we denote them with subscripts, such as
A1 , A2 , A3 , , A10 for a collection of 10 sets. For a family of sets indexed by
the natural numbers, we would denote the sets by A1 , A2 , A3 ,
∞
{ Ak }k =1
or
{ Ak }k∈N
where N = {1, 2,3,...} .
or maybe
Indexing sets in this way has
advantages. The reader will recall from calculus the notation for an infinite
sums and products to be
∞
∞
∑a
k
= a1 + a2 + ,
k =1
∏a
k
= a1a2 k =1
In a similar way, one defines the union and intersection of a family of
A1 , A2 , A3 , as follows.
Definition : Unions and Intersections of Families
Families of Sets The union of an
infinite number of sets A1 , A2 , in some universe U is defined to be
n
A1 ∪ A2 ∪ ∪ An = ∪ Ak = { x ∈U : x ∈ Ak for some k = 1, 2,..., n}
k =1
The intersection of the sets A1 , A2 , is
n
A1 ∩ A2 ∩ A3 ∩ An = ∩ Ak = { x ∈U : x ∈ Ak for all k = 1, 2, , n
}
k =1
Sometimes one deals with a more general families of sets
{ Aα }α∈Λ
where the
index α on the set ranges over some index set Λ ( Λ could be a finite set,
the natural numbers, or all values in an interval on the real line.). In this case
the union of the family { Aα }α ∈Λ is
∪ Aα = { x ∈U : x ∈ Aα
α ∈Λ
for at least one α ∈ Λ
}
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Section 2.2
Families of Sets
and the intersection of the family { Aα } is
∩ Aα = { x ∈U : x ∈ Aα
for all α ∈ Λ
}
α ∈Λ
The following examples illustrate these ideas.
Example 1
Let

Ak =  x : 0 ≤ x ≤

k 
 k = 1, 2,...
k + 1
Draw these sets and the find the following intersections.
a)
c)
Solution
The sets
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∪ Ak
k =1
3
∩A
k =1 k
A1, A2 , A3 ,
b)
d)
∞
∪ Ak
k =1
∞
∩A
k =1 k
are a family of monotomically increasing closed
intervals A1 ⊆ A2 ⊆ , each member in the family a subset of the next.
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Section 2.2
Families of Sets
Figure 1
3
a)
∪ A = [ 0,3 / 4]
k =1 k
b)
∪ A = [ 0,1)
k =1 k
∞
The right-endpoints of Ak get closer and closer
to 1 but are always less than 1: hence 1 is not
in the infinite union.
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c)
∩ A = [ 0,1/ 2 )
k =1 k
d)
∩ A = [ 0,1/ 2 )
k =1 k
∞
Example 2
Define the sequence of sets

1
Ak =  x ∈ :0 < x ≤ 1 −  , k = 1, 2,...
k

Find the union and intersection
∞
∞
∪ Ak and
∩A
k =2
k =2
k
.
Solution
Writing out a few sets, we see that by choosing k sufficiently large,
every real number in the interval ( 0,1) will belong in at least one of the
intervals Ak .
Also, the only points that belong to all of the intervals Ak are


1
2
the numbers in the interval  0,  . Hence
∞
1  1  2  3

A
=
∪
∪
k
 0,1 −  =  0,  ∪  0,  ∪  0,  ∪ = ( 0,1)
k   2  3  4
k =2
k =2 
∞
∞
1  1  2  3

 1
Ak = ∩  0,1 −  =  0,  ∩  0,  ∩  0,  ∩ =  0, 
∩
k   2  3  4
 2
k =2
k =2 
∞
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Section 2.2
Families of Sets
Margin Note:
▪ individual elements of sets are denoted by small letters x, y, a, b,...
▪ sets are denoted capital letters A, B, C ,...
▪ families of sets are denoted by script letters , , , …
Example 3 (Indexed Family) If An = {n + 1, n + 2, , 2n} for each natural
number n ∈ N , then
∞
∪ A = {n ∈ N : n ≥ 2}
k
k =1
∞
∩A
k
=∅
k =1
Example 4 (Indexed
(Indexed Family)
Find the following infinite intersections of the open
( −α , α )
and closed
intervals [ −α , α ] where the index α ranges over the real numbers .
∩ ( −α , α )
∩ [ −α , α ] .
α ∈
α ∈
Solution
In both cases for any x ≠ 0 there exists a real number α ∈ sufficiently
large so that x ∉ Aα . Thus any x ≠ 0 will not lie in either intersection. But
x = 0 belongs to both ( −α , α ) and [ −α , α ] for any real number α ∈ . Hence,
we conclude 0 belongs to both intersections, and so
∩ ( −α ,α ) = {0}
∩ [ −α , α ] = {0}
α ∈
α ∈
Example 5 (Set
(Set Projection)
Let S ⊆ × =
{( x, y ) : x, y ∈ }
plane and define the infinite family
be an arbitrary subset of the Cartesian
{ Ax }x∈
of sets
Ax = { y ∈ : ( x, y ) ∈ S }
( x, y ) ∈ S .
The
of this family is the projection of the set S onto the y -axis.
See
which represents for arbitrary x ∈ the y values such that
union
∪A
x
x∈
Figure 2.
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Section 2.2
Families of Sets
Projection of a Set
Figure 2
Extended Laws for Sets
Many laws of intersection and union of sets can be extended to families
of sets. We leave most of these proofs left to the reader.
Laws for Families of Sets
a)


A ∩  ∪ Bα  = ∪ ( A ∩ Bα )
 α ∈Λ  α ∈Λ


b) A ∪  ∩ Bα  = ∩ ( A ∪ Bα )
 α ∈Λ  α ∈Λ
c)
∩ Aα = α∪ Aα
α ∈Λ
d)
(DeMorgan’s Law)
∈Λ
∪ Aα = α∩ Aα
α ∈Λ
(DeMorgan’s Law)
∈Λ
Proof of d) Let
x ∈ ∪ Aα
α ∈Λ
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Section 2.2
Hence x ∉
∪ Aα
Families of Sets
and so x ∉ Aα for any α ∈ Λ . Thus x ∈ Aα for all α ∈ Λ and
α ∈Λ
thus x ∈
∩ Aα .
Hence
α ∈Λ
∪ Aα ⊆ α∩ Aα .
α ∈Λ
∈Λ
The proof the other way follows along the same lines and is left to the reader.
Defining Topologies on a Set
A topology on a set is a family of subsets of a set that that allows for the
study of convergence in the set. The study of topology forms much of the
basis of many areas of modern mathematics, such as real and complex
analysis. The area of mathematics that studies topologies and topological
spaces is called topology.
The idea is to introduce a family J of subsets of a given universal set U
(the universe might be the real numbers, complex numbers, or even a family
sets These open
of functions), where the sets in the family J are called open sets.
sets act as neighborhoods of points in U , thus allowing in a general way for
the discussion of convergence of elements in U . This family of open sets J
is called a topology on (or for) the universal set U . So is any collection of
subsets of U called a topology? The answer is no. There are three
restrictions on a family of subsets in order that they form a topology on U .
They are:
Definition: A topology J on a set U is any collection of subsets of U that
satisfies the following conditions:
(1) The empty set ∅ and U belong to the family J .
(2) The union of any collection of sets in J also belongs to J .
(3) The intersection of any two (or equivalently a finite number) sets in J
is also a member of J .
The sets in the topology J are called the open sets in the topology (or just
open sets).
sets Properties (2) and (3) say that the family J is closed under
unions and finite intersections.
intersections
As an illustration consider the universal set U of only three points
U = {a, b, c}
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Section 2.2
Families of Sets
So what subsets of U can we choose as our open sets? (i.e. sets in J ) We
know U has a total of 23 = 8 subsets. Below we list five families of subsets
of U , each of which satisfies the three given conditions for being a topology
on U .
The reader can verify (See Problem 7) each family J1 , J 2 , J 3 , J 4 , J 5
satisfies the conditions for being a topology on U . The topology J1 contains
two open sets, the empty set ∅ and the universe U , and is called the
indiscrete topology for U . At the other extreme the topology J 5 contains all
subsets of U is the discrete topology,
topology which means that every subset of U is
an open set. The other topologies J 2 , J 3 , J 4 are in between the two extreme
topologies.
J1 = {∅,U }
indiscrete topology
J 2 = {∅, {a} , U }
J 3 = {∅, {a} , {b, c} ,U }
J 4 = {∅, {a, b} ,U }
J 5 = {∅, {a} , {b} , {c} , {a, b} , {a, c} , {b, c} ,U }
discrete topology
Theorem 1 The family
J = {∅, {a} , {b, c} ,U }
is a topology on the set U = {a, b, c} .
Proof: We verify the three conditions required for a topology. The first condition is
verified since the topology J contains both the empty set ∅ and the universe U . The
second condition (closure under unions) also holds since one can check that the union any
combination of sets in J is also a member of J For example
{a} ∪ {b, c} = {a, b, c} ∈ J
∅ ∪ {a} = {a} ∈ J
{a} ∪ {a, b, c} = {a, b, c} ∈ J
Thirdly, it is also easy to see that the intersection of any two sets in J is also a member
of J . For example
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Section 2.2
.
Families of Sets
{a} ∩ {b, c} = ∅ ∈ J
{a} ∩ {b, c} = {a} ∈ J
{b, c} ∩ {a, b, c} = {b, c} ∈ J
In other words, J is closed under finite intersections.
So what do these topologies on U have to do with convergence? Wait till we get
to the study of analysis in Chapter 5 where we introduce the concepts of neighborhoods.
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Section 2.2
Families of Sets
Problems
A1 = {1, 2 } , A2 = {2,3 } , A3 = {3, 4}
1. Let
and in general
explicitly the following sets.
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a)
∪A
k
k =1
b)
∪A
k
k∈
c)
∪A
k
k ≥5
d)
∪
Ak
1≤ k ≤ 4
5
e)
∩A
k
k =1
f)
∩A
k
k∈
g)
∩
Ak
1≤ k ≤5
5
h)
∩A
k
k =1
i)
∩A
k
k∈
2.
For the following indexed families of sets find
∞
∪ Ak
∞
and
k =1
a)
Ak = [ 0,1 − 1/ k ]
b)
Ak = [ −1/ k ,1/ k ]
c)
Ak = ( 0,1/ k )
d)
Ak = {k} ∪ [1/ k , 2k ]
e)
Ak = [ k , k + 1]
f)
Ak = [ 0,1 + 1/ k ]
g)
Ak = {( x, y ) : x 2 + y 2 ≤ k}
∩A
k
k =1
.
Ak = {k , k + 1 } . Write
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Section 2.2
Families of Sets
(Algebra
(Algebra of Sets) A family Y of subsets of a universe U is called an algebra1
of sets if
3.
(i )
( ii )
A ∪ B is in Y whenever A and B are in Y
A is in Y whenever A is in Y
If this happens we say the family Y is closed under unions and complementation
complementation.
mplementation
Which of the following families of subsets of U = {a, b, c} are an algebra of
subsets on U .
a) P (U )
b)
c)
d)
{∅,U }
{∅, {a} ,U }
{∅, {a} , {b, c} ,U }
4. (Sets Indexed by Two Indices) Let Ai , j be a family of sets where i, j belong to
the index sets I , J respectively. A simple example might be the collection of
rectangles
Ai , j = {( x, y ) : 0 ≤ x ≤ i, 0 ≤ y ≤ j}
in the plane.
a)
Prove
∪∩ A
i, j
i∈I j∈J
⊆ ∩∪ Ai , j
j∈J i∈I
b) Find a counter example showing that the two sets in a) are not equal.
5.
(Sets of Length Zero) A subset A of the real numbers is said to have length
zero (called measure zero)
zero if ∀ε > 0 there exists a sequence Ak = ( ak , bk ) of
intervals such that they “cover” A : i.e.
∞
A ⊆ ∪ ( ak , bk )
k =1
1
Algebras of sets and sigma algebras (families of sets closed under countable unions) are fundamental to
the study of measure theory. Note the difference between an algebra of subsets and a topology of subsets
on a universe; just a minor difference makes for vastly different structures on the universe.
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Section 2.2
Families of Sets
∞
and their total length is less than ε ; that is
∑b
k
− ak < ε . Show that any
k =1
sequence of real numbers A = {ck } has measure zero. Hint: Cover each
element ck in the sequence by an interval ( ak , bk ) of length bk − ak = ε / 2k .
6. (Compact Sets) A subset A of the real numbers is said to be compact if
for every collection ℑ = ( a, b ) of open intervals that contains (or covers) A ;
{
}
i.e.
A⊆
∪ ( a, b )
( a ,b )∈ℑ
there exists a finite subcollection of ℑ that also contains (covers) A . Show
the set A = ( 0,1) is not compact by showing the following.
∞
a)
A is covered by
∪ ( 0,1 − 1/ k )
k =1
b) There does not exist a finite subcollection of ℑ whose union
contains A .
7. (Topologies) Verify that the following families are topologies for {a, b, c} .
J1 = {∅,U }
indiscrete topology
J 2 = {∅, {a} , U }
J 3 = {∅, {a} , {b, c} ,U }
J 4 = {∅, {a, b} ,U }
J 5 = {∅, {a} , {b} , {c} , {a, b} , {a, c} , {b, c} ,U }
discrete topology
8.. (Topologies) Which of the following families of {a, b, c} are topologies on
{a, b, c}
?
{
}
J = {∅, {a, b} , {b, c} , {a, b, c}}
J = {∅, {a} , {b, c}}
J = {∅, {b} , {b, c} , {a, b, c}}
a) J = ∅, {b} , {c} , {a, b, c}
b)
c)
d)