Basic Set Operations to Know for Intro to the Theory of Sets

Basic set operations are essential for understanding how sets interact. Key operations include union, intersection, complement, and difference. These concepts help us analyze relationships between sets, forming the foundation for more complex ideas in set theory.

1. Union

   • The union of two sets A and B, denoted as A ∪ B, includes all elements that are in A, in B, or in both.
   • It combines the elements of both sets without duplication.
   • The union operation is commutative (A ∪ B = B ∪ A) and associative ((A ∪ B) ∪ C = A ∪ (B ∪ C)).

2. Intersection

   • The intersection of two sets A and B, denoted as A ∩ B, consists of elements that are common to both A and B.
   • It is the set of elements that belong to both sets.
   • The intersection operation is also commutative (A ∩ B = B ∩ A) and associative ((A ∩ B) ∩ C = A ∩ (B ∩ C)).

3. Complement

   • The complement of a set A, denoted as A', includes all elements not in A, relative to a universal set U.
   • It represents everything outside of set A.
   • The complement operation is important for understanding set differences and is denoted as A' = U - A.

4. Difference

   • The difference between two sets A and B, denoted as A - B, includes elements that are in A but not in B.
   • It highlights what is unique to set A.
   • The difference operation is not commutative (A - B ≠ B - A).

5. Symmetric Difference

   • The symmetric difference of two sets A and B, denoted as A Δ B, includes elements that are in either A or B but not in both.
   • It can be expressed as (A - B) ∪ (B - A).
   • The symmetric difference is commutative (A Δ B = B Δ A) and associative ((A Δ B) Δ C = A Δ (B Δ C)).

6. Cartesian Product

   • The Cartesian product of two sets A and B, denoted as A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
   • It represents all possible combinations of elements from both sets.
   • The Cartesian product is not commutative (A × B ≠ B × A).

7. Power Set

   • The power set of a set A, denoted as P(A), is the set of all possible subsets of A, including the empty set and A itself.
   • If A has n elements, the power set will have 2^n elements.
   • The power set is fundamental in understanding the structure of sets and their relationships.

8. Subset and Superset

   • A set A is a subset of set B, denoted as A ⊆ B, if all elements of A are also in B.
   • Conversely, B is a superset of A, denoted as B ⊇ A.
   • The concepts of subset and superset are crucial for comparing sets and understanding their hierarchies.

9. Disjoint Sets

   • Two sets A and B are disjoint if they have no elements in common, denoted as A ∩ B = ∅.
   • Disjoint sets are important in probability and statistics, as they represent mutually exclusive events.
   • Understanding disjoint sets helps in analyzing relationships between different sets.

10. De Morgan's Laws

    • De Morgan's Laws describe the relationship between union and intersection through complements:
      • (A ∪ B)' = A' ∩ B'
      • (A ∩ B)' = A' ∪ B'.
    • These laws are essential for simplifying expressions involving sets and their complements.
    • They provide a foundational understanding of how operations on sets interact with each other.