Chapter 6: Sets and Counting

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Chapter 6 Sets and Counting

• The following sections will be covered
• 6.2 Sets
• 6.3 Basic Counting Principle
• 6.4 Permutations and Combinations

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Section 6.2 Sets

• A set is a collection of objects specified in
  such a way that we can tell whether any given
  object is or is not in the collection. Capital
  letters, such as, A, B, and C , are often used to
  designate particular sets. Each object in a set
  is called a member, or element, of the set.
• Symbolically,
• 
  

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Section 6.2 Sets

• A set without any elements is called the empty,
  or null, set. For example, the set of all people
  over 20 feet is an empty set.

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Section 6.2 Sets

• A set is usually described either by listing its
  elements between braces or by enclosing a
  rule within braces that determines the elements
  of the set. Thus,
• If P(x) is a statement about x, then

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Section 6.2 Sets

• Example Representing Sets
• (a) x x is a weekend daySaturday, Sunday
• (b) x x5105
• (c) x x is an odd positive integer1, 3, 5,

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Section 6.2 Sets

• If each element of a set A is also an element of
  the set B, we say A is a subset of B. If a set A
  and set B have exactly the same elements, then
  the two sets are said to be equal. Symbolically

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Section 6.2 Sets

• Remark
• Example Given A0,2,4,6, B0,1,2,3,4,5,6
• C2,6,0,4
• Indicate whether the following relationships are
  true or false

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Section 6.2 Sets

• Set Operations
• The union of sets A and B, denoted by
• A U B, is the set of all elements formed by
  combining all the elements of A and all elements
  of B into one set.
• The intersection of sets A and B, denoted by
• A B, is the set of all elements of A that
  are also in B.

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Section 6.2 Sets

• Venn diagrams

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Section 6.2 Sets

• Homework/Class Exercises
• Section 6.2 (page 390)
• 13, 14, 16, 19, 25. 26, 29-40, 59-70

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Section 6.3 Basic Counting Principles

• Three techniques Addition Principle, Venn
  Diagrams and Multiplication
• Addition Principle
• If A and B are any two sets, then

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Section 6.3 Basic Counting Principles

• Example According to a survey of business firms
  in a certain city, 750 firms offer their
  employees health insurance, 640 offer dental
  insurance, 280 offer health insurance and dental
  insurance. How many firms offer their employees
  health insurance or dental insurance?

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Section 6.3 Basic Counting Principles

• Solution Suppose Hthe set of firms that offer
  their employees health insurance, and let Dthe
  set of firms that offer their employees dental
  insurance, then

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Section 6.3 Basic Counting Principles

• Venn Diagrams
• Example A small town has two radio stations, an
  AM station and an FM station. A survey of 100
  residents of the town produced the following
  results, in the last 30 days, 65 people have
  listened to the AM station, 45 have listened to
  FM station, and 30 have listened to both
  stations.
• (A) How many people in the survey have listened
  to the AM station, but not to the FM station, in
  30-day period?

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Section 6.3 Basic Counting Principles

• (B) How many have listened to the FM station, but
  not to the AM station?
• (C) How many have not listened to ether station?
• (D) Organize this information in a table.

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Section 6.3 Basic Counting Principles

• Solution
• Let U be the group of people surveyed, let A be
  the set of people who have listened to the AM
  station and let F be the set of people who have
  listened to the FM station. Since U contains all
  the elements under consideration, it is a
  universal set for this problem. The complement of
  A, denoted A , is the set people in the survey
  group U who have not listened to the AM station.

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Section 6.3 Basic Counting Principles

• Similarly, F is the set people in the survey
  group U who have not listened to the FM station.
  Then

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Section 6.3 Basic Counting Principles

• Multiplication principle

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Section 6.3 Basic Counting Principles

• Example How many 4-letter code words are
  possible using the first 10 letters of alphabet
  if
• (A) No letter can be repeated?
• (B) Letters can be repeated?
• (C) Adjacent letters cannot be alike?

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Section 6.3 Basic Counting Principles

• Homework/Class Exercises
• Section 6.3 (page 399-340)
• 1-15 All, 21-29 Odd,35-39 Odd, 52-53 All

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Section 6.4 Permutations and Combinations

• Main topics Factorials, Permutations,
  Combinations, and Applications.
• Definition Factorial

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Section 6.4 Permutations and Combinations

• Question In how many ways can you arrange the 3
  objects in the set A, B,C without repetition?
• Answer
• List all of them. Answer ABC, ACB, BAC, BAC,
  CAB, CBA
• Remark If the arrangement of the elements didnt
  matter, then there would have been only one
  possibility ABC

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Section 6.4 Permutations and Combinations

• Question In how many ways can you arrange the 2
  objects in the set A, B,C, D without
  repetition?
• Answer
• List all of them. Answer AB, BA, AC, CA, AD, DA,
  BC, CB, BD, DB, CD, DC
• Remark If the arrangement of the elements didnt
  matter, then there would have been only 6
  possibilities AB, AC, AD, BC, BD, CD

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Section 6.4 Permutations and Combinations

• A permutation of a set of n distinct objects
  taken r at a time without repetition is an
  arrangement of r objects in a specific order.
• A combination of a set of n distinct objects
  taken r at a time without repetition is an
  r-element subset of the n objects. The
  arrangement of elements in the subset does not
  matter.

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Section 6.4 Permutations and Combinations

• Result
• The number of permutations of n distinct objects
  taken r at a time without repetition is given by
• The number of combinations of n distinct objects
  taken r at a time without repetition is given by

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Section 6.4 Permutations and Combinations

• Example From a committee of 12 people,
• In how many ways can you choose a chairperson, a
  vice-chairperson, a secretary, and a treasurer,
  assuming that one person cannot hold more than
  one position?
• In how many ways can we choose a subcommittee of
  4 people?

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Section 6.4 Permutations and Combinations

• Homework/Class Exercises
• Section 6.4 (pp. 412-413)
• 33, 34, 37, 38, 39, 47, 49, 50, 55, 57, 59, 63

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Next Class

• In our next class, we will learn section 6.4.
• Enjoy you homework, and have a wonderful day !