Set Concepts

• A set is a collection of objects.
• The objects in a set are called elements or members.
  • The integers 1 through 10 inclusive are a set
    • 1,2,3,4,5,6,7,8,9,10 are the elements.
  • The days of the week are a set
    • Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday are the elements.
• A set is well defined if the contents can be determined.
  • The set of integers between 1 and 10 inclusive is well defined.
  • The set of all grains of sand is a well defined set.
  • The set of the five best teachers at Edinboro is not well defined.
• There are three ways to describe a set
  • Descriptive format
    • Let A be the set of integers between 1 and 5 inclusive
    • Let B be the set of days of the week when this class meets
    • Let N be the set of integers greater than or equal to one.
    • Let C be the set of integers between 1 and 100 inclusive
    • Let D be the set of negative integers.
  • Roster format: list all of the members.
    • A = { 1,2,3,4,5}
    • B = {Monday, Tuesday, Wednesday, Thursday}
    • N = { 1, 2, 3, ...}
    • C = { 1, 2, 3, ..., 100}
    • D = { ..., -3, -2, -1}
  • Set Builder Notation
    • This is a formal notation for describing a set
    • X = {x | some algebraic expression involving set membership}
    • A = { a | a ∈ I and 1 ≤ a ≤ 5}
  • Some notation
    • The symbol ∈ means is an element of
    • The symbol ∉ means is not an element of
    • If A = { 1,2,3,4,5}
      • 1 ∈ A
      • 3.5 ∉ A
  • Two sets A and B are equal if all a ∈ A, a ∈ B and all b ∈ B, b ∈ A.
    • A = { 1 ,2 3, 4}, B = { 4, 3, 2, 1}, C = { a, b, c, d}
    • A = B
    • B = A
    • A ≠ C
    • B ≠ C
  • The cardinal number of a set n(A) is the number of elements in set a.
    • A = { 1 ,2 3, 4}, B = { 4, 3, 2, 1}, C = { a, b, c, d}
    • n(A) = 4
    • n(B) = 4
    • n(C) = 4
  • A set A is finite if n(A) is a natural number or 0.
    • A = { 1 ,2 3, 4}, N = {1, 2, 3, ...}
    • A is finite
    • B is infinte
  • A set A is equivalent to a set B if n(A) = n(B)
    • A = { 1 ,2 3, 4}, B = { 4, 3, 2, 1}, C = { a, b, c, d}
    • A, B and C are all equivalent.
  • The empty set contains no elements
    • {}
    • ∅
    • BUT NOT {∅} and not {{}}
  • The Universal Set is a set of all elements for any specific discussion.