Chapter 2.1 Set Concepts

• Sets are collections of objects.
• The objects in a set are called elements or members.
  • Let S be the set of all states in the united states.
  • Pennsylvania is an element of S.
• A set is well defined if the contents can be clearly determined.
  • Let A be the set of all students signed up for this class.
  • The set A is well defined, we can check to see if a student is signed up for the class or not.
  • Let N be the set of the five nicest days so far this year
  • Since nicest is not defined, determining which five days are the nicest. So N is not well defined.
• Sets can be defined three ways
  • The description format allows the st to be described.
    • The three sets above were given in description format.
    • This is most useful when the set is easy to describe in a well defined manner.
  • Roster format requires the elements of the set to be listed or partially listed.
    • P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
    • S = {Alabama, Alaska, Arizona, ... Wyoming}
    • Three dots ore ellipsis can be used to indicate continuing a pattern.
    • A = {2, 3, ... 9} is the set of integers between 1 and 10.
    • B = {1, 2, 3, ... 10} is the set of integers between 1 and 10 inclusive.
    • N = { 1, 2, 3, ...}
    • These are the natural numbers
    • In this case, the ellipsis indicate the set continues forever
    • I = { ... -3, -2, -1, 0, 1, 2, 3, ...}
    • Is the set of integers.
  • Set-builder Notation is a very formal method for writing sets
    • D = { x | 0 ≤ x < 10}
    • Let d be the set of x such that x is greater or equal to 0 and less than 10.
• ∈ is the membership symbol
  • A = { 1, 2, 3, 4, 5}
  • 5 ∈ A
  • Five is an element of A.
  • 9 ∉ A
  • Nine is not an element of A.
• For a set A, n(A) is the number of elements in the set.
  • This is known as the cardinal number
  • P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
  • n(P) = 8
  • A = {1, 2, 3, 4, 5 }
  • n(A) = 5
• A set is finite if the number of elements is 0 or a natural number
  • Let A be the set of students signed up for this class. A is finite because it is either 39 (the 1:00 class) or 22 (the 2:00 class).
• A set is infinite if the number of elements is not 0 or a natural number.
  • The set N = {1, 2, 3, ... } is infinite
• Caution, when giving examples of finite and infinite sets, please select clear examples for which the answer is known.
  • Let S be the set of stars in the universe.
• Two sets (A and B) are equal if every element of A is also an element of B and every element of B is also an element of A.
  • V = { 9, 7, 5, 3, 1}
  • W = { 1, 3, 5, 7, 9}
  • X = { 1, 2, 3, 4, 5}
  • Y = { paper, scissors, rock, lizard, spock}
  • W and V are equal (W=V)
  • W and X are not equal (W ≠ X)
• Two sets are (A and B) are equivalent if n(A) = n(B).
  • All of the sets above are equivalent.
• The empty set is a set which contains no elements.
  • ∅
  • {}
  • n(∅) = 0
  • n({}) = 0
  • { ∅} is not the empty set.
  • { {} } is not the empty set.
• The Universal Set (U) is the set of all elements under consideration.