Sets, Whole Numbers, and Numeration

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Chapter 2

• Sets, Whole Numbers, and Numeration

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2.1 Sets as a Basis for Whole Numbers

• Definition A set is a collection of objects,
  and the individual objects are called elements or
  members of the set.
• Notation
• Listing Set Builder

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Two Special Kinds of Sets

• Definition The set containing no elements is
  called the empty set or the null set.
• Definition The universal set is the set of all
  elements under consideration in a given
  discussion. The letter U is reserved for the
  universal set.

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Comparing Sets

• Definition Two sets A and B are equal if they
  have exactly the same members.
• Notation
• If A is not equal to B, we write .
• Note Order is not a requirement for equality.

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Comparing Sets

• Definition A 1-1 correspondence between two sets
  A and B is a pairing of the elements of A with
  the elements of B so that each element of A
  corresponds to exactly one element of B, and vice
  versa.
• A and B are equivalent.

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The Idea of Subset

• Definition The set A is a subset of set C if
  every element of A is also an element of C.
• Notation
• If A is not a subset of C, then we write
  .
• Special Cases
• If A is any set, .
• The empty set is a subset of all sets.

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Why Two Different Notations?

• Think about less than or equal to and strictly
  less than.
• Definition The set A is a proper subset of the
  set C if and .
• Notation

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Venn Diagrams
A
U
C
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Listing All Possible Subsets
Find all subsets of the set
Size of Subset Subsets of this size of subsets of this size
0 1
1 3
2 3
3 1
Total 8
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Set Operations

• Definition The union of sets A and B is the set
  of elements that are members of either A or B or
  both A and B.
• Notation

A
B
U
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Set Operations

• Definition The intersection of sets A and B is
  the set of all elements common to both A and B.
• Notation

A
B
U
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Set Operations

• If , then A and B are said to be
  disjoint.

A
B
U
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Set Complement

• Definition If A is a subset of the universal
  set U, the complement of A is the set of elements
  in U that are not in A.
• Notation

A
U
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Set Difference

• Definition The difference of sets A and B is the
  set of elements in A that are not in B.
• Notation

A
B
U
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Cartesian Product of Sets

• Definition The Cartesian product of set A with
  set B, written A X B, is the set of all ordered
  pairs (a,b) where and

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Order of Operations

1. Parentheses
2. Complement
3. Union, intersection, set difference from left to
   right.

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DeMorgans Laws

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Survey Problems

• Start with your Venn Diagram, and work from the
  inner most region outward.

A
B
A
B
U
U
C
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Counting elements in a set

• Definition The number of elements in a set A is
  called the cardinal number of set A.
• Notation
• A set is finite if its cardinal number is a whole
  number.
• An infinite set is one that is not finite.

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2.3 The Hindu-Arabic System

• Hindu-Arabic Numeration System A.D 800
• Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 These 10
  symbols, or digits, can be used in combination to
  represent all possible numbers.
• Decimal System Grouping in sets of 10 is a basic
  principal of this system. Our system is a base
  10 system.

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• Hindu-Arabic Numeration System contd
• Positional System Each of the various places in
  the number 6523, for example, has its own
  value
• Additive and Multiplicative The value of a
  Hindu-Arabic numeral is found by multiplying each
  place value by its corresponding digit and then
  adding all the resulting products.

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Non-Decimal Base Systems

• Suppose we had only one hand, so instead of
  having 10 digits we only had five. What would
  that mean for our numeration system?
• Base five instead of base 10.
• Digits 0, 1, 2, 3, 4

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2.4 Relations and Functions

• Relations are used in mathematics to represent a
  relationship between two numbers or objects.
• Relations can be illustrated in different ways.
• Arrow Diagrams
• Set of ordered pairs
• Algebraic Equation

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Properties of Relations

• A relation may have three useful properties
• Reflexive Property A relation R on a set A is
  reflexive if for all
• Symmetric Property A relation R on a set A is
  symmetric if whenever then also.
• Transitive Property A relation R on a set A is
  transitive if whenever and
  then .

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Functions

• Definition A function is a relation that
  matches each element of a first set to an element
  of a second set in such a way that no element in
  the first set is assigned two different elements
  in the second set.NOTATION
• The first set is called the domain, and the
  second set is called the codomain. The set of
  all elements in the codomain that the function
  pairs with an element in the domain is called the
  range.

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Sequences

• Definition An arithmetic sequence is a sequence
  in which successive terms differ by the same
  number. This difference is called the common
  difference.
• Definition A geometric sequence is a sequence
  in which each term after the first can be found
  by multiplying by the same number. This
  multiplier is called the common ratio.

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Generator Functions

• To find the nth term of an arithmetic sequence,
  use the following formula
• To find the nth term of an geometric sequence,
  use the following formula