Set Concepts

• Some definitions
  • A set is a collection of objects.
  • The objects in a set are called elements or members
• The trees on campus form a set. Each individual tree is an element.
• The types of trees of campus form a different set. Each type of tree (Sugar Maple, Scots Pine, ...) is an element.
• A set is well defined if the contents of the set can be determined.
  • The people signed up for this class is well defined.
  • The three best people in this class is not. We have various methods for determining best.
• Sets are generally defined in three different ways
  • Description Method: The above sets were described in this method:
  • Roster Form: List all elements of the set
    • The set of all integers between 3 and 7
    • { 4, 5, 6}
    • The set of all integers between 3 and 7 inclusive
    • { 3, 4, 5, 6, 7, }
    • The set of natural numbers, or counting numbers
    • N = { 1, 2, 3, ...}
  • Set builder notation
    • S = { x | condition}
    • Read S is the set of all x such that condition
    • N = {x | x ∈N and 3< x < 7}
    • P is the set of all planets in our solar system
    • P = {p | p is a planet}
  • A set is finite if it contains no elements, or the number of elements is a natural number.
  • • The set of all planets is a finite set.
  • A set is infinite if it is not finite.
    • The set of natural numbers is an infinite set.
    • So are the integers, reals, ...
  • Two sets A and B are equal if they contain the same elements.
    • { 1,2,3} and {1, 2, 3} are equal
    • { 1, 3, 5, 7, 9} and {1, 2, 3, 4, 5} are not equal.
  • The cardinality of a set is the number of elements in a set.
    • A = { 1, 3, 5, 7} , n(A) = 4
  • Two sets are equivalent if they contain the same number of elements.
    • { 1, 3, 5, 7, 9} and {1, 2, 3, 4, 5} are equivalent.
  • A set containing no elements is the empty set
    • {} or ∅
    • {∅} is not the empty set, it contains ∅
    • {{}} is not the empty set, it contains {}
  • The Universal set is the set that contains all of the elements for a discussion.
    • If we are talking about this class, U = {s | s is a student in this class}
  • Examples: 13-18, 19-24, 27 (but 10 to 17), 32, 43, 46, 67-70, 79, 80
• Subsets
  • A set A is a subset of set B, written A ⊆ B if and only if all of the elements in A are also in B.
    • A = { 1, 2, 3}, B= { 1, 2, 3, 4, 5}, A ⊆ B
    • A = { a, b, c}, B= { 1, 2, 3, 4, 5}, A ⊄ B note, html does not have the proper symbol, this should be ⊆ with a slash through it
  • A is a proper subset of B, written A⊂ B, if and only if each element of A is in B, but A ≠ B
    • A = { 1, 2, 3}, B= { 1, 2, 3, 4, 5}, A ⊂ B
    • A = { 1, 2, 3, 4, 5}, B= { 1, 2, 3, 4, 5}, A ⊄ B
  • The empty set is a subset of every set, including itself.
  • Not the difference between an element (∈ and a subset ⊂)
    • {a} ∉ { a, b, c}
    • {a} ⊂ { a, b, c}
    • a ∈ { a, b, c}
    • a ⊄ { a, b, c}
  • If a set A contains n elements, there are 2ⁿ subsets of A.
    • A = {}, n(A) = 0, 2⁰=1 subsets: {}
    • A = { x}, n(A) =1, 2¹ = 2 subsets: {}, {x}
    • A = {I, II}, n(A) =2, 2² = 4 subsets: {}, {I}, {II}, {I, II}
  • Examples: 7, 9, 11, 13, 28, 30, 31, 38, 51