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