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• Set


  Any well-defined collection, group, list, aggregation, or class of distinct objects
• Well-defined


  The membership in the set should be clear
• Distinct


  Each object is different from the other objects within the set, such that in cases of duplication, only one occurrence of the objects is listed in the set
• Elements or Members


  Objects in the set. Examples are articles, people, conditions, places, letters or numbers
• Examples of Well-Defined sets

  • The set of bachelor degrees offered in the Bulacan State University
  • The set of letters of the Filipino alphabet
  • The set of odd numbers less than 100
  • The set of numbers 1,3,5,7,9 and 11
  • A collection of old coins
• Set Notation


  Uppercase letters A,B,C,...,X,Y,Z
• Elements/Members


  Lowercase letters a,b,c,...,x,y,z and are enclosed in braces { }
• a ∈ A


  a is an element of set A
• a ∉ A


  a is not an element of set A
• Two Methods of Writing Sets


  1. By Enumeration or Roster Method

  2. By Defining or Rule Method
• Enumeration or Roster Method


  The elements of a set are listed or enumerated (in any order but without repetition) and enclosed in braces. Each element is separated from the others by commas.
• Examples of Enumeration or Roster Method


  • A is the set of all even natural numbers between 20 and 40
  • C is the set of subjects offered in the 2nd semester for 1st year IT students
  • Set V is the set of consonants in the word "STRUCTURES"
• Defining or Rule Method


  Instead of listing the member of the set, the rule method is used to define the elements of the set. The "set builder notation" x|P(x)}
• Examples of Defining or Rule Method


  • A = {x|x is an even natural number between 20 and 40}
  • C = {x|x is a course offered in the CICT}
  • V = {x|x is a consonant in the word "STRUCTURES"}
• The set builder notation is an efficient notation for describing sets
• The symbol " : " can also be used instead of the vertical bar
• Other letters can also be used instead of "x" such as v, w, y, z, etc.
• Kinds of Sets

  • Finite Set
  • Infinite Set
  • Unit Set / Singleton Set
  • Empty Set / Null Set / Void Set
  • Universal Set
  • Subsets, Superset & Proper Subset
  • Equal Sets
  • Equivalent Sets
  • Disjoint Sets / Non-Intersection
  • Complementary Sets
  • Power Sets
  • Product Sets
  • Classes of Sets
• Finite Set


  A set with k distinct elements, where k ∈ N. The given elements are limited or countable and its last element can be identified.
• Examples of Finite Sets

  • A = {x|x is a positive integer less than 17}
  • C = {x|x is an odd integer between 500 and 1000}
• Infinite Set


  A set whose elements are unlimited or uncountable.
• Examples of Infinite Sets


  • D = {0,1,2,3,...}
  • E = {x|x is a negative odd integer}
  • F = {x|x is a proper fraction}
• Unit Set / Singleton Set


  A set with only ONE ELEMENT.
• Examples of Unit/Singleton Sets


  • G = {x|x is an integer greater than 10 but less than 12}
  • H = {x|x is a number between 68 and 70}
• Empty Set / Null Set / Void Set


  A set containing NO OBJECTS or ELEMENTS. The braces with no elements { } or the symbol " ∅ " are used to denote and empty set.
• Examples of Empty/Null/Void Sets

  • I = {x|x is an integer greater than 0 but less than -1}
  • J = {x|x is an even number between 7 and 8}
• Universal Set


  Set of all possible elements under consideration. Universal sets are denoted by the uppercase letter "U"
• Examples of Universal Sets


  • U = {a, b, c, ..., x, y, z}
  • U = {1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 5.0}
• Subset


  A is a subset of B denoted by A ⊆ B, if every element of A is in B.
• Superset


  If A is a subset of B, then we can say that B is a superset of A, and write B ⊇ A.
• Proper Subset


  A is a proper subset of B, denoted by A ⊂ B if A ⊆ B and A ≠ B.
• Examples of Subsets, Supersets, and Proper Subsets


  • A = {1,2,3}; B = {4,3,2,1}; C = {1,2,3,4,5,6}

  A ⊆ B, A ⊂ B, A ⊆ C, and A ⊂ C

  B ⊇ A

  C ⊇ A

  ∅ ⊆ A, ∅ ⊆ B and ∅ ⊆ C
• Equal Sets


  A = B, if and only if A ⊆ B AND B ⊆ A. The symbol "≠" is used to denote if two sets are not equal.
• Examples of Equal Sets


  • {1,2,3,4} = {4,3,2,1}

  {2,3,4} = {an integer between 1 and 5}
• Equivalent Sets


  Two sets A and B are equivalent if they have exactly the same number of elements. The symbol for set equivalence is ≈
• Examples of Equivalent Sets


  • A = {1,2,3}; B = {a,b,c}; C = {μ,ε,α}

  A ≈ B, A ≈ C, and B ≈ C
• Disjoint Sets / Non-Intersection


  Two sets A and B are disjoint if they have no elements in common.
• Examples of Disjoint Sets


  • The sets {3,7,9,18,21} and {1,17,4} are disjoint.

  The set of positive numbers is disjoint from the set of negative numbers.

  The set of numbers less than 100 and the set of numbers greater than 100 are disjoint.
• Complementary Sets


  Two sets are complementary sets if they are disjoint and if when combined collectively they form the Universal set. The notation for the complement of A is A^C.
• Examples of Complementary Sets


  • Let U = {0,1,2,3,...,9} and A = {0,2,4,6,8}. Then A^C = {1,3,5,7,9}