Section 2'6: Infinite Sets

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Section 2.6 Infinite Sets

• Dr. Fred Butler
• Math 121 Fall 2004

2
Review of One-to-One Correspondences

• Recall that a one-to-one correspondence between
  two sets A and B is a way of pairing up every
  element in A with exactly one element in B.
• For example
• S Maryland, Pennsylvania, West Virginia
• C Charleston, Annapolis, Harrisburg

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Proper Subsets Review

• Also recall that the set B is a proper subset of
  the set A if and only if all the elements of the
  set B are elements of the set A, and the set B ?
  A.
• If A 1, 3, 5, 7, 9, 11, then
• B 11, 3, 7
• is a proper subset, but
• C 3, 1, 9, 11, 7, 5
• is not a proper subset (because C A).

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Infinite Sets

• An infinite set is a set that can be placed in
  one-to-one correspondence with a proper subset of
  itself.
• Note the importance of the word proper in the
  above definition.
• If it werent in the definition, we could use the
  fact that any set A is a subset of itself, and we
  could easily give a one-to-one correspondence
  between A and itself.
• In this case, every set would be infinite!

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The Set of Natural Numbers is Infinite

• Consider the set N1,2,3,4, of natural
  numbers.
• Let S2,3,4,5,, which is a proper subset of N
  (since 1 is not in S).
• N 1, 2, 3, 4, 5, 6,
• S 2, 3, 4, 5, 6, 7,
• The general way of pairing elements in N with
  elements in S is n?n1.
• This shows that N is infinite.

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The General Term of an Infinite Set

• The general term of a set is a formula involving
  n, such that when 1 is substituted for n we get
  the first term of the set, when 2 is substituted
  for n we get the second term of the set, etc.
• When showing that the set A is infinite, we need
  to demonstrate a one-to-one correspondence
  between A and a proper subset of A by showing how
  to pair the general terms of the two sets.

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An Example of Finding the General Term

• Consider the set
• 4, 9, 14, 19, .
• How can we find a formula for the general term of
  this set?
• In 4, 9, 14, 19, , we note that
• 1. 9 4 5
• 2. 14 9 5
• 3. 19 14 5
• etc.
• The difference between each successive term is 5.
• This means that the general term looks like 5na
  for some number a.

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An Example of Finding the General Term contd.

• The general term looks like 5na for some a.
• The first term is 4, so for n1 we must have 4
  5(1) a.
• Thus a -1, and the general term is 5n1.
• So our list looks like
• 4, 9, 14, 19, , 5n-1,

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Class Question 1.7

• What is the general term of the set
• 4, 6, 8, 10, 12, 14, 16, ?
• 1. 2n 2. 2n2 3. 2n4

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The Set of Even Numbers is Infinite

• Consider the set
• E 2, 4, 6, 8, , 2n,
• of even numbers.
• Consider the proper subset
• S 4, 6, 8, 10, , 2n2,
• obtained by removing 2 from E.

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The Set of Even Numbers is Infinite contd.

• Then the desired one-to-one correspondence is
  given by
• E 2, 4, 6, 8, , 2n,
• S 4, 6, 8, 10, , 2n2,
  ,
• where we pair the general term 2n in E with the
  general term 2n2 in S.
• This correspondence shows E is infinite.

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Countable Sets

• A set is countable if it is finite, or if it can
  be placed in one-to-one correspondence with the
  set of counting numbers.
• For example, 1, 2, 5, 6, 8, a, is countable
  because it is finite.
• So is the empty set Ø.
• N 1, 2, 3, 4, 5, 6, is also countable.

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Another Example of a Countable Set

• We will now show that the set of the odd counting
  numbers
• O 1, 3, 5, 7, 9, 11,
• is countable, by finding a one-to-one
  correspondence between O and N.
• Using the methods described earlier, we can see
  that the general term of O is
• 2n 1, because
• 2(1) 11, 2(2) 13, 2(3) 15, etc.

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An Example of a Countable Set contd.

• So we have the one-to-one correspondence
• N 1, 2, 3, 4, 5, , n,
• O 1, 3, 5, 7, 9, , 2n1,
  
• which in general sends n to 2n1.
• We have established a one-to-one correspondence
  between N and O, which shows that O is countable.
  

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The Cardinal Number ?0

• Any infinite set that can be placed in one-to-one
  correspondence with the counting numbers has
  cardinal number ?0, pronounced aleph-null.
• Said another way, any countable infinite set has
  cardinal number ?0.
• This symbol is made up of ?, the first letter of
  the Hebrew alphabet, with a 0 subscript.
• This definition immediately gives us that the
  cardinal number of the set N of counting numbers
  is ?0.

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The Cardinal Number of the Odd Counting Numbers

• In establishing a one-to one correspondence
  between the odd counting numbers
• O 1, 3, 5, 7, 9, 11,
• and the counting numbers
• N 1, 2, 3, 4, 5, 6, ,
• we have also shown that the set O has cardinal
  number ?0.

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Other Sets with Cardinality ?0

• It can be shown (although we will not do so) that
  the set of integers
• Z , -3, -2, -1, 0, 1, 2, 3, ,
• and the set of rational numbers
• Q p/q p e Z, q e Z, and q ? 0
• (the set of all fractions), also both have
  cardinality ?0.
• These sets both seem to be much bigger than the
  set of counting numbers N, but in terms of
  cardinal numbers they are the same size!

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Cardinal Numbers and Unions

• Recall the formula for finite sets A and B
• n(A U B) n(A) n(B) n(A n B).
• Recall that two sets A and B are disjoint if they
  have no elements in common (in which case n(A n
  B) 0).
• So when A and B are finite disjoint sets, the
  above formula becomes
• n(A U B) n(A) n(B).

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Cardinal Numbers and Unions contd.

• If A1,2,3 and B7,8,9,10,11, then A and B
  are disjoint,
• A U B 1,2,3,7,8,9,10,11,
• and
• n(A U B) 8 3 5 n(A) n(B).
• However, if A1,2,3,4 and B2,4,6,8,10, then
  A and B are not disjoint,
• A U B1,2,3,4,6,8,10,
• and
• n(A U B) 7 ? n(A) n(B).

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Cardinal Numbers and Unions contd.

• The formula
• n(A U B) n(A) n(B)
• for A and B finite disjoint sets is saying that
  the cardinal number of the union of A and B is
  just the sum of the cardinal numbers of A and B.
• We will now see that strange things happen when
  we try to apply this formula to infinite sets.

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A Paradox

• Earlier, we showed that the cardinal number of
  the set of odd counting numbers O1,3,5,7,9,
  is ?0.
• In a similar fashion, we can show that the
  cardinal number of the set of even counting
  numbers E2,4,6,8,10, is also ?0.
• Finally, we know that the cardinal number of the
  set of counting numbers N1,2,3,4,5, is ?0.

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A Paradox contd.

• We have that O1,3,5,7,9, and E2,4,6,8,10,
  are disjoint sets, and
• O U E 1,3,5,7,9, U 2,4,6,8,10
• 1,2,3,4,5,6, N.
• Applying the formula
• n(N) n(OUE) n(O) n(E),
• we get that
• ?0 ?0 ?0 !

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An Infinite Set with Cardinality Bigger than ?0

• The set R of real numbers includes the counting
  numbers, integers, rational numbers, irrational
  numbers (things like v2), and even transcendental
  numbers (things like p).
• Georg Cantor, the mathematician responsible for
  all of this work on infinite sets, proved that R
  has cardinal number bigger than ?0.
• We will discuss irrational numbers, p, and the
  set of real numbers R later in the course.

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Lecture Summary

• An infinite set is a set that can be placed in
  one-to-one correspondence with a proper subset of
  itself.
• We need to demonstrate such a one-to-one
  correspondence by showing how to pair the general
  terms of the two sets.
• A set is countable if it is finite, or if it can
  be placed in one-to-one correspondence with the
  set of counting numbers.

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Lecture Summary contd.

• An infinite set that can be placed in one-to-one
  correspondence with the counting numbers have
  cardinal number ?0, pronounced aleph-null.
• The set R of real numbers is an infinite set that
  has cardinality bigger than ?0 .

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Homework

• Do problems from Section 2.6 of the textbook.
• HW Quiz 1 is due Wednesday, Sep. 8, in Web CT by
  1100 PM.
• Study for Unit 1 Exam, held Thursday Sep. 09 and
  Friday Sep. 10.