2.1 Set Concepts

• A set is a collection of objects.
• The objects are called elements or members
• We will use sets in the probability section.
• We will solve practical set problems.
• We need some terminology describing sets.
• A set is well defined if it's contents can be clearly determined.
  • Is the set of the five richest people well defined? Yes
  • Is the set of the five best business people well defined? No
• A set is finite if the number of elements can be counted.
• If a set is not finite, it is infinite
• Are the following sets finite?
  • The set of all even numbers - no
  • The set of all people on earth - yes
• Three ways to write a set
  • Description: what we have been using so far
  • Roster Method
    • List all of the elements, inside a pair of braces {}, separated by commas.
      • The set of instructors for this class I={Dan}
      • The set of grades given in this class G={A,B+,B,C+,C,D+,D,F,i}
      • The set of integers greater than seven and less than 10. C={8,9}
      • Natural numbers or counting numbers N = { 1,2,3,4, ...}
      • We use ... to indicate continuation.
  • Set Builder Notation
    • D = { x | x condition}
    • D is the set of x such that x meets some condition}
    • A = { x | x is an integer and 3 ≤ x ≤ 8}
    • The set of days in the week that start with an s
    • S = { x | x is a day of the week and the first letter of x is an S}
    • The set of integer between 1 and 100
    • E = { x | 1 ≤ x ≤ 100}
    • The set of multiples of five
    • F = { x | x ∈ N, x = 4k for k ∈ N}
  • In the last example, ∈ means an element is in a set.
  • 3 ∈ {1,3,5,7,9}
  • 4 ∉ {1, 3, 5, 7, 9}
  • If A is a set n(A) is the number of elements in the set.
  • n({1,3,5,7,9}) = 5
  • Two sets A and B, are equal (written A=B) if they contain the same elements.
  • Two sets A and B, are equivalent if n(A) = n(B)
  • The empty set {} or ∅ is a set with no elements.
  • The universal set or U is a set that contains all elements related to a specific discussion.

2.2 Subsets