Elementary Set Theory

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Elementary Set Theory
Definitions Set Operations Venn Diagrams De
Morgans Laws
2
Terminology

• A set is a collection of objects
• The objects are called the elements, members or
  points of a set and is denoted by
• If then
• Equality of sets, A B
• Empty or Null Set, or
• Subset, read A is contained in B
• Power Set (set of all subsets)
• Universal Set

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Venn Diagrams and Set Operations

• The idea of using shaded circles to illustrate
  relationships between sets to attributed to
  British logician John Venn (1834-1923).
• Here is a Venn diagram illustrating the
  relationship between the sets A and B.

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Intersection of Sets
Key Word

• The intersection of sets A and B is the set of
  all items that are in both A AND B.
• We express intersection, A?B.

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Unions of sets
Key Word

• The union of sets A and B is the set of all items
  that are either in A OR B.
• We express union as A?B.
• In math, the word or should be interpreted as A
  or B or both A and B.

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Disjoint Sets

• Two events are or disjoint if A?B?, that is,
  they have no elements in common.

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Complements

• A complement of a set A is everything that is in
  the universal set, U, but not in the set A.
• The complement
• is denoted, Ac.

AC
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De Morgans Laws

• If A and B are any two sets

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Repetition of Objects

• Note that repetitions of objects may be
  eliminated from a set, for example

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Example 1

• An investigation of a number of automobile
  accidents revealed the following information
• 18 accidents involved alcohol and excessive speed
• 26 accidents involved alcohol
• 12 accidents involved excessive speed but not
  alcohol
• 21 accidents involved neither alcohol nor
  excessive speed
• How many accidents were investigated?

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Example 2

• A new car dealership offers three options to its
  customers moon-roof, satellite radio, and a
  diesel engine. The dealership listed the
  following information in its yearly tax records
• 200 cars sold
• 75 without any options
• 10 with all three options
• 40 included satellite radio and diesel engine
• 30 moon-roof and diesel engine
• 20 moon and satellite radio
• 80 moon-roof
• 60 satellite radio
• 70 diesel engine
• Explain why the Bureau of Internal Revenue
  ordered a complete audit of the dealerships
  records.

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Example 3

• Shade the indicated sets on a Venn diagram
• a.
• b.

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Review

• Union of two sets, A ? B, represents all items
  that are in set A OR set B
• Intersection of two sets, A ? B, represents all
  items that are in both A AND B
• The complement of set A, Ac, is everything that
  is in the universal set but not in the set A
• Two sets are disjoint if A ? B ?

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Exercises

• Let A and B be any two subsets of the universal
  set, U. What does
• (A ? Ac) equal?
• (A ? Ac) equal?
• If A and B are disjoint, what does
• (A ? B) equal?