Mathematical Sets: Definitions, Formulas, and Applications

1. Definitions and Set Notation

A set is a well-defined collection of distinct objects, called elements. Sets are typically denoted by capital letters, with elements enclosed in curly braces:

A = {1, 2, 3, 4}   # A set containing four numeric elements

Ways to represent sets:

• Listing elements: Explicit enumeration, e.g. {a, b, c}
• Descriptive notation: Using a property, e.g. B = {x | x is an even number less than 10} → {2, 4, 6, 8}
• Algebraic rules: Expressing sets via formulas

2. Types of Sets

• Empty Set (∅): Contains no elements: C = {} or C = ∅
• Finite and Infinite Sets: Finite: limited elements; Infinite: e.g., N = {1, 2, 3, ...}
• Universal Set (U): The set of all elements under consideration
• Complement of a Set (Aᶜ): Elements in U that are not in A. If A = {2, 4, 6}, U = {1, 2, 3, 4, 5, 6}, then Aᶜ = {1, 3, 5}

3. Set Operations

• Union (A ∪ B): All elements from both sets

  A = {1, 2, 3}, B = {3, 4, 5}  
  A ∪ B = {1, 2, 3, 4, 5}
  

• Intersection (A ∩ B): Common elements

  A ∩ B = {3}
  

• Difference (A − B): Elements in A not in B

  A − B = {1, 2}, B − A = {4, 5}
  

• Complement (Aᶜ): Elements not in A with respect to U

  U = {1, 2, 3, 4, 5}, A = {1, 2}  
  Aᶜ = {3, 4, 5}
  

• Cartesian Product (A × B): All ordered pairs

  A = {x, y}, B = {1, 2}  
  A × B = {(x,1), (x,2), (y,1), (y,2)}
  

4. Essential Set Theory Formulas

• Distributive Law:

  A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  

• Element Count in Union:

  |A ∪ B| = |A| + |B| − |A ∩ B|
  

• Cartesian Product of Multiple Sets:

  |A₁ × A₂ × ... × Aₙ| = |A₁| × |A₂| × ... × |Aₙ|
  

5. Applications of Sets

• Number Theory: Classifying and organizing sets like natural numbers, integers, primes
• Algebra & Probability: Defining events, sample spaces
• Computer Science: Data models, search queries, database logic
• Mathematical Logic: Set-based representation of statements and truth values
• Graph Theory: Modeling connections and network edges as sets