SET THEORY

• Set


  An unordered collection of distinct objects, called elements or members of the set. A set is said to contain its elements.
• It is common for sets to be denoted using uppercase letters. Lowercase letters are usually used to denote elements of sets.
• Set definition


  A set can be defined by simply listing its members inside curly braces.
• Membership


  Denoted using the ∈ symbol, as in 4 ∈ A
• Non-membership


  Denoted using the ∉ symbol, as in S ∉ A
• Important sets in discrete mathematics

  • N = {0, 1, 2, 3, ...} (natural numbers)
  • Z = {..., -2, -1, 0, 1, 2, ...} (integers)
  • Z+ = {1, 2, 3, ...} (positive integers)
  • Q = {p/q | p in Z, q in Z, and q ≠ 0} (rational numbers)
  • R (real numbers)
  • R+ (positive real numbers)
  • C (complex numbers)
• Cardinality


  A measure of a set's size, meaning the number of elements in the set.
• Descriptive Form


  Elements of sets are stated in words.
• Set-Builder Form


  Characterize all those elements in the set by stating the property or properties they must have to be members.
• Roster Form


  Listing all the elements of the set, separated by commas and enclosed within curly braces {}.
• Descriptive Form


  • The set of all vowels in the English alphabet
• Set-Builder Form


  • {x: x is a vowel in the English alphabet}
• Roster Form


  • {a, e, i, o, u}
• Descriptive Form


  • The set of all odd positive integers less than or equal to 15
• Set-Builder Form


  • {x: x is an odd number and 0 < x ≤ 15}
• Roster Form


  • {1, 3, 5, 7, 9, 11, 13, 15}
• Descriptive Form


  • The set of all positive cube numbers less than 100
• Set-Builder Form

  • {x: x is a cube number and 0 ≤ x < 100}
• Roster Form


  • {1, 8, 27, 64}
• Empty/Null Set


  A set which does not contain any element. It is denoted by ∅ or {}.
• Singleton Set


  A set which contains only one element.
• Finite Set


  A set which contains a definite number of elements.
• Finite Sets

  • Set of days of the week

  Set of alphabets in English

  Set of regions in the Philippines
• Infinite Set


  A set in which all elements cannot be listed, i.e. a set containing never-ending elements. Ellipsis (...) is used to indicate that the elements of the set goes on.
• Infinite Sets

  • Set of natural numbers: N = {0, 1, 2, 3, ...}

  Set of negative integers: Z = {..., -3, -2, -1}
• Equivalent Sets


  Two sets are said to be equivalent if they have the same number of elements. The symbol for denoting an equivalent set is ↔.
• Equivalent Sets

  • A = {1, 2, 3}

  B = {a, b, c}

  |A| = 3; |B| = 3

  A ↔ B
• Equal Sets


  Two sets are said to be equal if and only if they contain the same elements for all x (x in A ⇔ x in B).
• Equal Sets

  • A = {blue, yellow, red}

  B = {red, blue, yellow}

  A = B
• Subset


  Set A is a subset of set B if and only if every element of A is also an element of B for all x (x in A → x in B).
• Subset

  • A = {2, 4, 6, 8, 10}

  B = {6, 4, 8, 10, 2}

  A ⊆ B
• Proper Subset


  Set A is a proper subset of set B if every element of A is also an element of B, but they are not equal sets. A ⊂ B but A ≠ B.
• Proper Subset

  • A = {a, c, e}

  B = {a, b, c, d, e}

  A ⊂ B
• Power Set


  Given the set A, the power set of A is the set of all subsets of the set A, denoted as P(A) or 2^A.
• Universal Set


  The set of all elements under consideration, denoted by capital U. All other sets are subsets of the universal set.
• Universal Set

  • A = {1, 3, 5, t}

  B = {c, a, t, 3}

  U = {1, 3, 5, c, a, t}
• Union


  The union of sets A and B, denoted A ∪ B, is the set that contains those elements that are either in A or in B, or in both.
• Union

  • K = {a, b}

  L = {c, d}

  M = {b, d}

  K ∪ L = {a, b, c, d}

  K ∪ M = {a, b, d}
• Union

  • C = Set of primary colors

  D = Set of secondary colors

  C ∪ D = {red, blue, yellow, green, orange, violet}
• Intersection


  The intersection of sets A and B, denoted A ∩ B, is the set of all elements that are common to both A and B.