Thinking Mathematically

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Thinking Mathematically

• Chapter 2 Set Theory
• 2.1 Basic Set Concepts

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Basic Set Concepts

• A set is a collection of objects. Each object is
  called an element of the set.
• A set must be well defined
• Its contents can be clearly determined
• Its clear if an object is or is not a member of
  the set.

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Representing Sets

• Word Description Describe the set in your own
  words, but be specific so the elements are
  clearly defined.
• Roster Method List each element, separated by
  commas, in braces.
• Set-Builder Notation x x is word
  description.

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The Set of Natural Numbers

• N 1,2,3,4,5,
• This is an example of a set
• We will be talking a lot more about sets of
  numbers in Chapter 5

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Examples Representing Sets

• Exercise Set 2.1 3, 5, 13, 15, 25
• Well defined sets (T/F)
• The five worst U.S. presidents
• The natural numbers greater than one million
• Write a description for the set
• 6, 7, 8, 9, , 20
• Express this set using the roster method
• The set of four seasons in a year.
• x x ? N and x gt 5

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The Empty Set

• The empty set, also called the null set, is the
  set that contains no elements.
• The empty set is represented by
• or Ø

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Examples Empty Sets

• Exercise Set 2.1 35, 37, 41, 45
• Which sets are empty
• x x is a women who served as U.S. president
  before 2000
• x x is the number of women who served as U.S.
  president before 2000
• x x lt2 and x gt 5
• x x is a number less that 2 or greater than 5

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The Notation ? and ?

• The symbol ? is used to indicate that an object
  is an element of a set. The symbol ? is used to
  replace the words is an element of
• The symbol ? is used to indicate that an object
  is not an element of a set. The symbol ? is used
  to replace the words is not an element of

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Example Set elements

• Exercise Set 2.1 51, 59, 63 (T/F)
• 5 ? 2, 4, 6, , 20
• 13 ? x x ? N and x lt 13
• 3 ?3, 4

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Definition of a Sets Cardinal Number

• The cardinal number of set A, represented by
  n(A), is the number of distinct elements in set
  A. The symbol n(A) is read n of A.
• Repeated elements are not counted.

Exercise Set 2.1 71 C x x is a day of the
week that begins with the letter A n( C) ?
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Definition of a Finite Set

• Set A is a finite set if n(A) 0 or n(A) is a
  natural number. A set that is not finite is
  called an infinite set.

Exercise Set 2.1 91 x x ? N and x gt
100 Finite or infinite?
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Definition of Equality of Sets

• Set A is equal to set B means that set A and set
  B contain exactly the same elements, regardless
  of order or possible repetition of elements. We
  symbolize the equality of sets A and B using the
  statement A B.

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Definition of Equivalent Sets

• Set A is equivalent to set B means that set A
  and set B contain the same number of elements.
  For equivalent sets, n(A) n(B).

Exercise Set 2.1 85 A 1, 1, 1, 2, 2, 3, 4 B
4, 3, 2, 1 Are these sets equal? Are these
sets equivalent?
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Thinking Mathematically

• Chapter 2 Set Theory
• 2.3 Venn Diagrams and Set Operations
• well come back to 2.2

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Definition of a Universal Set
A universal set, symbolized by U, is a set that
contains all of the elements being considered in
a given discussion or problem.
Exercise Set 2.3 3 A Pepsi, Sprite B Coca
Cola, Seven-Up Describe a universal set that
includes all elements in sets A and B
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Venn Diagrams
Disjoint sets have no elements in common.
All elements of B are also elements of A.
The sets A and B have some common elements.
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Definition of the Complement of a Set
The complement of set A, symbolized by A, is the
set of all elements in the universal set that are
not in A. This idea can be expressed in
set-builder notation as follows A x x ?
U and x ? A .
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Complement of a Set
U
A
A
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Example Set Complement

• Exercise Set 2.3 11
• U 1, 2, 3,, 20
• A 1, 2, 3, 4, 5
• B 6, 7, 8, 9
• C 1, 3, 5, , 19
• D 2, 4, 6, , 20
• C ?

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Definition of Intersection of Sets

• The intersection of sets A and B, written A?B,
  is the set of elements common to both set A and
  set B. This definition can be expressed in set
  builder notation as follows
• A ? B x x ? A AND x ? B

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Definition of the Union of Sets

• The union of sets A and B, written A ? B, is the
  set of elements that are members of set A or of
  set B or of both sets. This definition can be
  expressed in set-builder notation as follows
• A ? B x x ? A OR x ? B

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The Empty Set in Intersection and Union

• For any set A
• 1. A n ? ?
• 2. A ? ? A

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Examples Union / Intersection

• Exercise Set 2.3 17, 19, 33, 35
• U 1, 2, 3, 4, 5, 6, 7
• A 1, 3, 5, 7
• B 1, 2, 3
• C 2, 3, 4, 5, 6
• A ? B ?
• A ? B ?
• A ? ? ?
• A n ? ?

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Cardinal Number of the Union of Two Sets

• n(A U B) n(A) n(B) n(A nB)
• Exercise Set 2.3 93
• Set A 17 elements
• Set B 20 elements
• There are 6 elements common to the two sets
• How many elements in the union?

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Thinking Mathematically

• Chapter 2 Set Theory
• 2.2 Subsets

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Definition of a Subset of a Set

• Set B is a subset of set A, expressed as
• B ? A
• if every element in set B is also an element in
  set A.

Every set is a subset of itself A ? A
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Definition of a Proper Subset of a Set

• Set B is a proper subset of set A, expressed as B
  ? A, if set B is a subset of set A and sets A
  and B are not equal ( A ? B ).
• What is an improper subset?

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The Empty Set as a Subset

1. For any set B, ? ? B.
2. For any set B other than the empty set, ? ? B.

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Example Subsets

• Exercise Set 2.2 3, 45, 43, 47
• -3, 0, 3 ____ -3, -1, 1, 3
• (?, ?, both, neither)
• Ralph ? Ralph, Alice, Trixie, Norton (T/F)
• Ralph ? Ralph, Alice, Trixie, Norton (T/F)
• ? ? Archie, Edith, Mike, Gloria (T/F)

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Thinking Mathematically

• Chapter 2 Set Theory
• 2.4 Set Operations and Venn Diagrams With Three
  Sets

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Example Operations with three sets Exercise
Set 2.4 3, 15

• U 1, 2, 3, 4, 5, 6, 7
• A 1, 3, 5, 7
• B 1, 2, 3
• C 2, 3, 4, 5, 6
• (A ? B) n (A ? C)

• U a, b, c, d, e, f, g, h
• A a, g, h
• B b, h, h
• C b, c, d, e, f
• (A ? B) n (A ? C)

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Example Venn Diagrams

• Exercise Set 2.4 35, 37

U
A
B
4,5
10, 11
1, 2, 3
6
7, 8
9
12
C
13
(A ? B) ?
A ? B ?
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Example Venn Diagrams

• Exercise Set 2.4 27, 29

U
A
B
II
III
I
V
IV
VI
VII
C
A ? C ?
A n B ?
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De Morgans Laws(using Venn Diagrams as a proof)

• (A U B)' A' n B' The complement of the union
  of two sets is the intersection of the complement
  of those sets.

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De Morgans Laws

• (A n B)' A' U B' The complement of the
  intersection of two sets is the union of the
  complement of those sets.

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Examples DeMorgans Laws

• U 1, 2, 3, 4, 5, 6, 7
• A 1, 3, 5, 7
• B 1, 2, 3
• (A n B) ' ?
• A ' U B ' ?

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Thinking Mathematically

• Chapter 2 Set Theory