2.1, Set Concepts

• A set is a collection of objects.
• The objects in a set are called elements or members.
• A set is well defined if the contents can be clearly determined.
• Three ways to write sets.
  • Descriptive
  • Set builder notation
  • Roster form.
• Descriptive: give a description of the set.
  • Let W be the set of days of the week.
  • Let I be the set of instructors in this class.
  • Let R be the set of students in this class.
• Roster format:
  • W = { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
  • I = {Dan}
  • Sometimes we use three dots (or ellipsis) to show continuation.
  • N = { 1, 2, 3, 4, 5, ...} the natural numbers
  • I = { ... -3. -2, -1, 0, 1, 2, 3, ...}, the integers,
  • H = { 1, 2, 3, ... 100}, the natural numbers less than or equal to 100
• We use the symbol ∈ to show something is an element of a set..
  • 4 ∈ H
  • -2 ∉ N
• Set builder notation
  • S = { x | condition on X}
  • H = {x | x ∈ N and x ≤ 100}
• The cardinality of a set is the number of elements in the set.
  • n(S)
  • sometimes written |S|
• Finite verses infinite sets
  • If n(S) ∈ N, then S is finite.
  • If n(S) = 0, then S is finite.
  • Otherwise S is infinite.
• Two sets, A and B, are equal (written A=B) if and only if A and B contain exactly the same elements.
  • E = { 1, 2, 3}
  • F = { 3, 2, 1}
  • G = { A, B, C}
  • H = { a, b, c, d}
  • E = F
  • E ≠ G
• Two sets A and B are Equivalent if n(A) = n(B)
  • From above E, F and G are equivalent.
• The empty set contains no elements.
  • {}
  • ∅