Sections 2.1

• Some Definitions
  • A set is a collection of objects.
  • The objects in the set are called elements or members
    • A = {1, 2, 3}, A is a set, 1, 2, and 3 are elements of A
    • 2 ∈ A;
  • A set is well defined if all elements can be determined.
    • The set of people in this class is well defined.
    • The set of the best three student in this class is not.
  • The Natural or counting numbers are the integers starting at 1.
  • A set listed this is way is listed by description
  • N = { 1, 2, 3, ...}
  • A set listed this way is in roster format
  • The ... means continues this pattern.
  • Set Builder Notation is a way to list a set without enumerating all members.
    • X = { x | x ∈ N and x is even}
    • X is the set of all x such that x is in the natural numbers and x is even.
    • X = { 2, 4, 6, 8, ...}
  • By the way, 3 ∉ X
  • A set is infinite if it contains infinitely many elements.
  • A set is finite otherwise.
    • N is infinite.
    • Q = {q | q ∈ N and q < 100,000} is finite
  • The cardinality or cardinal number of a set is the number of elements.
    • n(Q) = 100,000
    • A = { 1, 2, 3, 5, 7, 9}, n(A) = 6
  • Two sets are equal (A=B) if every element of A is in B, and every element of B is in A.
  • The order of the elements in a set is not important.
    • A = { 1, 2, 3, 5, 7, 9}
    • B = { 9, 7, 5, 3, 2, 1}
    • A = B
  • Two sets are equivalent if they have the same cardinal number.
    • C = {a, b, c, e, g, i}
    • C is equivalent to A, since n(C) = n(A)
    • But C ≠ A
  • A set with no elements is called the NULL or Empty set.
  • Written {} or Ø
  • The Universal set is a set containing all elements relevant to a discussion.

Sections 2.2

• A is a subset of B (A⊆B) if all elements of A are also elements of B.
• Otherwise A⊄B
• If A⊆B but A≠B, then A is a proper subset of B.
• Written A⊂B
• The empty set is a subset of every set.
• A set S, has 2^n(S) subsets.
• {} has {} or 2⁰
• {a} has {} and {a} or 2¹ or 2 subsets
• {a, b} has {}, {a}, {b}, {a,b} or 4

Homework

• Page 50, 13-85 odd
• Page 58, 7 through 53 odd.