Set theory

• Set


  A collection of well-defined objects
• Examples of sets


  • Set of football players
  • Set of books
  • Set of chess
  • Set of students in a class
• Sets


  • Must be well-defined
• Types of sets


  • Finite sets
  • Infinite sets
  • Equivalent sets
  • Equal sets
  • Subsets
• Set-builder notation


  Representing sets using capital letters and listing the members
• Finite set


  A set with a definite beginning and end
• Finite sets


  • Set of English alphabets
  • Set of natural numbers less than 10
• Infinite set


  A set with no definite beginning or end
• Infinite set


  • Set of natural numbers
• Equivalent sets


  Sets with the same number of elements
• Equal sets


  Sets with the same members
• Subset


  A set that is contained within another set
• The number of subsets of a set with n elements is 2^n
• Unit set


  A set with only one member
• Universal set


  The set containing all the elements under consideration
• Intersection


  The set of elements common to two or more sets
• The notation A = {a1, a2, ...} represents the set with elements a1, a2, ...
• A set is any collection of objects, called elements or members
• If x is an element of a set A, we write it as x∈A
• The notation A = {x | P(x)} represents the set of x such that P(x) is true.
• The notation A = {x : P(x)} represents the same as above but is used when there are no quantifiers (either universal or existential)
• An element can be anything that has meaning within the context of the problem being considered.
• Sets are denoted by capital letters (e.g., A, B, C)
• The notation A = {x / P(x)} represents the same as above but is used when there are both quantifiers (universal and/or existential).
• Elements of a set are denoted by lowercase letters (e.g., x, y, z).
• If x is not an element of a set A, we write it as x∉A
• An empty set has no elements; its symbol is Ø (the Greek letter omega)
• The empty set (or null set), denoted by Ø, contains no elements at all
• Two sets are equal if they have exactly the same elements
• The universal set is the largest possible set; it contains every object being considered
• An intersection is the set of elements that are contained in both sets
• The empty set has no elements; its symbol is Ø or {}
• The power set of a set A is the set of all subsets of A
• Sets are denoted by capital letters like A, B, C, etc.
• Elements of sets are denoted by lowercase letters like a, b, c, etc.
• The cardinality of a finite set is the number of elements in the set
• Set


  A collection of well-defined, distinct and distinguishable objects or things
• Elements (members)


  Objects or things that are part of a set
• Specifying a set


  1. Listing the elements in the set (e.g. A = {1,2,3})

  2. Set builder notation (e.g. B = {x|2 < x < 5})