SETS

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• SETS

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Elements of a set

• The individual members of the set are called the
  elements of the set.
• is an element of a set of dishes.

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Elements of a set

• The symbol means is an element of
• (hint looks like an e for element!)
• means that 2 is an element of the
  set of numbers 1, 2, 3

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Elements of a set
Remember Ghostbusters?
means 4 is not an element of the set 1, 2, 3
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A well-defined set

• A set is well-defined if it has clear rules
  that make it obvious if something is an element
  of the set or not.

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A well-defined set?

• The set of the members of the 2007 Colts team is
  a well-defined set.

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George Bush a Colt?

• Even though hes in the picture, you know George
  Bush is not really a member of the team because
  the 2007 Colts team is a well-defined set.

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NOT a well-defined set

• The set of the 5 best Colts players in history is
  not a well-defined set because different people
  may have different opinions.

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Cardinality of a set

• The cardinality (or cardinal number) of a set is
  just the number of elements in the set.
• The cardinality of 2, 4, 6 is 3.

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Cardinality of a set

• The symbol for the cardinality (or cardinal
  number) of a set is n( ).
• If A 2, 4, 6 then n(A) 3.

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Equivalent sets

• Two sets are equivalent if they have the same
  number of elements or the same cardinality n( ).
• If A 2, 4, 6 and B 1, 3, 5
• n(A) 3 and n(B) 3.
• A and B are equivalent sets.

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Equal sets

• Two sets are equal if they have exactly the same
  elements.
• Order doesnt matter.
• If A 2, 4, 6 and B 6, 4, 2
• A and B are equal sets.

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Equal Sets are Equivalent

• If A 2, 4, 6 and B 6, 4, 2
• A and B are equal sets.
• n(A) 3 and n(B) 3
• A and B are equivalent sets

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Equivalent Sets may NOT be Equal

• If A 2, 4, 6 and B 1, 3, 5
• n(A) 3 and n(B) 3
• A and B are equivalent sets
• but A and B are NOT equal sets.

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Naming a set

• Sets are traditionally named with capital
  letters.
• Let N a, b, c, d, e

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Describing a set

• Set are typically enclosed in braces
• You can describe a set by using
• words
• a roster
• set-builder notation

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Describing a set

• The easiest way to describe many sets is by using
  words.
• B all blue-eyed blonds in this class
• There isnt a really good mathematical equivalent
  for that!

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The Roster Method

• Just like in gym class when they read the roster,
  this simply lists all of the object in the set.
• Team A Andrews, Baxter, Jones, Smith, Wylie

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When to use a roster

• A roster works fine if you only have a few
  elements in the set.
• However, many sets of numbers are infinite.
• Listing each member would take the rest of your
  life!

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When to use a roster

• If an infinite set of numbers has a recognizable
  pattern, you can still use a roster.
• First establish the pattern
• then use an ellipsis
• to indicate that the pattern goes on
  indefinitely.
• 1, 2, 3, 4

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The Natural Numbers

• The set of natural numbers N is also called the
  set of counting numbers.
• N 1, 2, 3,

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Finite Sets

• We think of a finite set as having a countable
  number of elements.
• Mathematically, that means that the cardinality
  of the set (number of elements) is a natural
  number 1, 2, 3,
• Geek Patrol If A is a finite set, n(A) N

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Set-builder notation

• For many infinite sets it is easier to just use a
  rule
• to describe the elements of the set.
• We use braces to let everyone know that
  were talking about a set.
• We use a vertical line to mean such that
• lets see
  how it works

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Set-builder notation

• x x gt 0
• The set of all x such that x is positive.
• We couldnt possible use a roster to describe
  this set because it includes fractions and
  decimals as well as the counting numbers.
• We couldnt set up the pattern to begin with!

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The Empty Set

• If a set has NO elements, we call it the empty
  set.
• The symbol that we will use for the empty set is
• The cardinal number for the empty set is 0.

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OOPS!

• Notice that the empty set is the one set that we
  do not enclose in braces.
• is not empty!
• It is a set with one element the Greek letter
  Phi.

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Vocabulary
Element Well-defined Cardinality Equivalent vs.
equal Finite vs. infinite Roster vs. set builder
notation Empty set