Introduction to Sets

• A set is a collection of objects.
• The objects are called elements or members
• Sets are a universal part of mathematics.
• A set is well defined if the contents can be clearly determined.
  • or if it can be determined if something is in the set or not.
  • Some well defined sets:
    • The set of presidents of the United States.
    • The set of students in this room.
    • The integers.
  • Some not well defined sets:
    • The best teachers at Edinboro.
• There are three ways to describe a set.
  • Descriptive format
    • Provide a name (normally upper case letter) and a description in English.
    • Let A be the set of letters in the alphabet.
    • Let N be the set of natural numbers.
    • Let S be the set of students in this room.
  • Roster format
    • Provide a name (upper case letter) and a list of elements.
    • Let P = { Clinton, Trump}
    • Let W = { Monday, Tuesday, Wednesday, Thursday}
    • If there is a pattern we can just list the first few, ellipses, and the last.
    • Let A = {a, b, c, ... z}
    • Or we can provide the pattern in an infinite set
    • Let N = {1, 2, 3, ...}
    • N is the set of natural numbers
    • Let I = { ..., -3, -2, -1, 0, 1, 2, 3, ...}
    • I is the set of integers
  • Set Builder Notation
    • A predefined pattern
    • Let X = { variable | condition on variable}
    • Let T = { t | t ∈ N and 0 < t < 10}
    • "Let T bet the set of t such that t is a natural number and t is between 0 and 10"
    • Let E = { e | e ∈ I and e/2 ∈ I}
    • Let E be the set of e such that e is an integer and e/2 is an integer.
  • On ranges
    • 0 < x < 10 is between 0 and 10
    • 0 ≤ x ≤ 10 is between 0 and 10 inclusive
• A set is finite if the number of elements in the set is 0 or a natural number.
  • A = { 1, 2, 3, 4} is finite
  • B = {} is the empty set and this is a finite set with no elements.
  • C = { 1, 2, 3, ... 50} is finite
  • D = {1, 2, 3, ...} is infinite.
  • If you are asked to give an example of an infinite set, use this one.
• Two sets A and B are equal if all elements of A are elements of B and all elements of B are elements of A.
• The cardinal number of a set is the number of elements in that set.
  • The cardinal number the set A is written n(A)
  • This was once |A|
  • Two sets A and B, are equivalent if n(A) = n(B)
• The universal set is the set of all objects under consideration at any time.
• Do exercises on page 48