DISCRETE MATH FINALS

• Set


  Fundamental concept that represents a collection of distinct objects, considered as an object in its own right
• Sets


  • Used to group related items together
  • Used to define relationships between objects
• Elements/Members


  The objects inside a set
• Universal Set


  The set that contains all the objects or elements under consideration in a particular context or discussion
• Empty Set


  The set that contains no elements. It is unique and a subset of every set
• Empty Set


  • Serves as the identity element for the union of sets
  • Serves as the null element for the intersection of sets
• Equal Sets


  Two sets A and B are equal if and only if every element of A is an element of B, and every element of B is an element of A
• A = B: ∀x (x∈A ⟺ x∈B)
• Subset


  Set A is a subset of Set B (written as A ⊆ B) if every element of A is also an element of B
• A ⊆ B means ∀x( x∈A → x∈B)
• Proper Subset


  Set A is a proper subset of Set B (written A ⊂ B) if A ⊆ B and A ≠ B
• Superset


  Set B is a superset of Set A (written B ⊇ A) if every element of A is also an element of B
• Proper Superset


  Set B is a proper superset of set A (written B ⊃ A) if B ⊇ A and B ≠ A
• Difference


  The difference of sets A and set B (written A − B or A ∖ B) is the set of elements that are in A but not in B
• A − B = {x ∣ x∈A and x∉B}
• Complement


  The complement of a set A (written A' or ) is the set of elements that are in the universal set U but not in A
• A' = {x ∣ x∈U and x∉A}
• Cardinality


  The number of elements in a set, a measure of the "size" of the set
• Finite Set


  A set with a finite number of elements, its cardinality is the count of those elements
• Infinite Set


  A set with an infinite number of elements, its cardinality can be described in terms of types of infinity (e.g., countable or uncountable)
• Power Set


  The set of all possible subsets of a set A, including the empty set and A itself
• P(A) = {∅, {1}, {2}, {1, 2}} if A = {1, 2}
• Tuple


  An ordered list of elements, used to describe sequences where order matters
• Cartesian Product


  The set of all ordered pairs (a, b) where the first element 'a' is from set A and the second element 'b' is from set B
• A × B = { (a, b) ∣ a∈A and b∈B}
• Roster Notation


  A way of specifying a set by listing its elements, separated by commas, and enclosed within curly braces {}
• Interval Notation


  A way of representing subsets of the real number line by specifying the endpoints of the intervals
• Types of Intervals


  • Closed Interval [a, b]
  • Open Interval (a, b)
  • Half-Open (or Half-Closed) Interval [a, b) or (a, b]
• Closed interval takes precedence when given set is in Roster Method
• Set-Builder Notation


  A concise way of specifying a set by describing the properties that its members must satisfy
• General form: { x ∣ 'condition'}
• The symbol | is read as "such that"

  
• Steps to write Set-Builder Notation

  1. Identify the elements in the given

  2. Look for patterns or properties

  3. Determine the general form

  4. Specify the Domain [if applicable]

  5. Write the Set-Builder Notation
• When the elements are multiples of a number or have mathematical properties, use the format: {x | 'formula' for n ∈ 'numbers multiplied'}