The Nature of Sets

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Section 2.1
Math in Our World

• The Nature of Sets

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Learning Objectives

• Define set.
• Write sets in two different ways.
• Classify sets as finite or infinite.
• Define the empty set.
• Find the cardinality of a set.
• Decide if two sets are equal or equivalent.

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Sets

• A set is a well-defined collection of objects.
• By well-defined we mean that given any object, we
  can definitely decide whether it is or is not in
  the set.
• Each object in a set is called an element or a
  member of the set.

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EXAMPLE 1 Listing the Elements in a Set

• Write the set of months of the year that begin
  with the letter M.
• SOLUTION
• The months that begin with M are March and May.
  So, the answer can be written in set notation as
• M March, May
• Each element in the set is listed within braces
  and is separated by a comma.

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Two methods are commonly used to designate a set

• Roster Form
• The elements of the set are listed between
  braces, with commas between the elements.
• Description
• This uses a short statement to describe the set.

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Sets are generally named with a capital letter.

• The Set of Natural Numbers (Counting Numbers) is
  listed
• N 1, 2, 3, 4
• The Integers
• I -3, -2, -1, 0, 1, 2, 3

The three dots, or an Ellipsis, indicates that
the pattern continues indefinitely.
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EXAMPLE 2 Writing Sets Using the Roster Method

• Use the roster method to do the following
• (a) Write the set of natural numbers less than 6.
• (b) Write the set of natural numbers greater than
  4.
• SOLUTION
• (a) 1, 2, 3, 4, 5
• (b) 5, 6, 7, 8, . . .

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Set Notation

• The symbol ? is used to show that an object is a
  member or element of a set. For example, let set
• A 2, 3, 5, 7, 11.
• Since 2 is a member of set A, it can be written
  as
• 2 ? 2, 3, 5, 7, 11 or 2 ? A
• Likewise, 5 ? 2, 3, 5, 7, 11 or 5 ? A
• When an object is not a member of a set, the
  symbol ? is used. Because 4 is not an element of
  set A, this fact is written as
• 4 ? 2, 3, 5, 7, 11 or 4 ? A

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EXAMPLE 3 Understanding Set Notation

• Decide whether each statement is true or false.
• (a) 27 ? 1, 5, 9, 13, 17, . . .
• (b) z ? v, w, x, y, z

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EXAMPLE 3 Understanding Set Notation

• SOLUTION
• (a) The pattern shows that each element is 4 more
  than the previous element. So the next three
  elements are 21, 25, and 29 this shows that 27
  is not in the set. The statement is false.
• (b) The letter z is an element of the set, so the
  statement is false.

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EXAMPLE 4 Describing a Set in Words

• Use the descriptive method to describe the set E
  containing 2, 4, 6, 8, . . . .
• SOLUTION
• The elements in the set are called the even
  natural numbers. The set E is the set of even
  natural numbers.

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EXAMPLE 5 Using Different Set Notations

• Designate the set S with elements 32, 33, 34, 35,
  using
• (a) The roster method.
• (b) The descriptive method.
• SOLUTION
• (a) 32, 33, 34, 35, . . .
• (b) The set S is the set of natural numbers
  greater than 31

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EXAMPLE 6 Writing a Set Using an Ellipsis

• Using the roster method, write the set containing
  all even natural numbers between 99 and 201.
• SOLUTION
• 100, 102, 104, . . . , 198, 200
• If a set contains many elements, we can again use
  an ellipsis to represent the missing elements.

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Finite and Infinite Sets

• If a set has no elements or a specific natural
  number of elements, then it is called a finite
  set. A set that is not a finite set is called an
  infinite set.
• The set p, q, r, s is called an finite set
  since it has four members p, q, r, and s. The
  set 10, 20, 30, . . . is called an infinite set
  since it has an unlimited number of elements the
  natural numbers that are multiples of 10.

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EXAMPLE 7 Classifying Sets as Finite or
Infinite

• Classify each set as finite or infinite.
• (a) Set R is the set of letters used to make
  Roman numerals.
• (b) 100, 102, 104, 106, . . .
• (c) Set M is the set of people in your immediate
  family.
• (d) Set S is the set of songs that can be written.

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EXAMPLE 7 Classifying Sets as Finite or
Infinite

• SOLUTION
• (a) The set is finite since the letters used are
  C, D, I, L, M, V, and X.
• (b) The set is infinite since it consists of an
  unlimited number of elements.
• (c) The set is finite since there is a specific
  number of people in your immediate family.
• (d) The set is infinite because an unlimited
  number of songs can be written.

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Empty Set or Null Set

• A set with no elements is called an empty set or
  null set. The symbols used to represent the null
  set are or ?.

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EXAMPLE 8 Identifying Empty Sets

• Which of the following sets are empty?
• (a) The set of woolly mammoth fossils in museums.
• (b) ?

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EXAMPLE 8 Identifying Empty Sets

• SOLUTION
• (a) There is certainly at least one woolly
  mammoth fossil in a museum somewhere, so the set
  is not empty..
• (b) Be careful! Each instance of and ?
  represents the empty set, but ? is a set with
  one element ?.

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Cardinal Number

• The cardinal number of a finite set is the number
  of elements in the set. For a set A the symbol
  for the cardinality is n(A), which is read as n
  of A.
• For example, the set R 2, 4, 6, 8, 10 has a
  cardinal number of 5 since it has 5 elements.
  This could also be stated by saying the
  cardinality of set R is 5.

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EXAMPLE 9 Finding the Cardinality of a Set

• Find the cardinal number of each set.
• (a) A 5, 10, 15, 20, 25, 30
• (b) B 10, 12, 14, . . . , 28, 30
• (c) C 16
• (d) ?

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EXAMPLE 9 Finding the Cardinality of a Set

• SOLUTION
• (a) n(A) 6 since set A has 6 elements
• (b) n(B) 11 since set B has 11 elements
• (c) n(C) 1 since set C has 1 element
• (d) n(?) 0 since there are no elements in an
  empty set

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Equal and Equivalent Sets

• Two sets A and B are equal (written A B) if
  they have exactly the same members or elements.
  Two finite sets A and B are said to be equivalent
  (written A ? B) if they have the same number of
  elements that is, n(A) n(B).

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EXAMPLE 10 Deciding if Sets are Equal or
Equivalent

• State whether each pair of sets is equal,
  equivalent, or neither.
• (a) p, q, r, s a, b, c, d
• (b) 8, 10, 12 12, 8, 10
• (c) 213 2, 1, 3
• (d) 1, 2, 10, 20 2, 1, 20, 11
• (e) even natural numbers less than 10
• 2, 4, 6, 8

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EXAMPLE 10 Deciding if Sets are Equal or
Equivalent

• SOLUTION
• (a) Equivalent
• (b) Equal and equivalent
• (c) Neither
• (d) Equivalent
• (e) Equal and equivalent

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Classwork

• p. 50-53 7, 9, 11, 15, 17, 21, 27, 35, 37, 49,
  51, 53, 57, 59, 53, 65, 67, 75, 79, 80, 83, 85,
  87, 91 (a, b), 96 (a, b), 98 (a, b)