Set Identities to Know for Intro to the Theory of Sets

Set identities are key rules that help us understand how to work with sets in a flexible and efficient way. These identities simplify operations like union and intersection, making it easier to manipulate and combine sets in various scenarios.

1. Commutative Laws

   • The order of elements in a union or intersection does not affect the result: A ∪ B = B ∪ A and A ∩ B = B ∩ A.
   • This law highlights the flexibility in combining sets, making it easier to manipulate expressions.
   • It applies to both finite and infinite sets, reinforcing the foundational nature of set operations.

2. Associative Laws

   • The grouping of elements in unions or intersections does not change the outcome: (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C).
   • This law allows for the simplification of complex set expressions by rearranging parentheses.
   • It emphasizes that the operations can be performed in any order without affecting the final result.

3. Distributive Laws

   • Union and intersection distribute over each other: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
   • This law is crucial for simplifying expressions and solving problems involving multiple sets.
   • It illustrates the relationship between different operations, providing a framework for combining sets.

4. Identity Laws

   • The union of a set with the empty set is the set itself: A ∪ ∅ = A, and the intersection of a set with the universal set is the set itself: A ∩ U = A.
   • These laws establish the foundational elements that do not change the identity of a set.
   • They are essential for understanding how sets interact with neutral elements in operations.

5. Complement Laws

   • The union of a set and its complement equals the universal set: A ∪ A' = U, and the intersection of a set and its complement equals the empty set: A ∩ A' = ∅.
   • These laws define the relationship between a set and everything outside of it.
   • They are fundamental in set theory, emphasizing the completeness of sets and their complements.

6. Idempotent Laws

   • A set unioned or intersected with itself remains unchanged: A ∪ A = A and A ∩ A = A.
   • This law simplifies expressions by eliminating redundant operations.
   • It reinforces the idea that repeating the same operation on a set does not alter its identity.

7. Domination Laws

   • The union of any set with the universal set is the universal set: A ∪ U = U, and the intersection of any set with the empty set is the empty set: A ∩ ∅ = ∅.
   • These laws illustrate the extremes of set operations, showing how certain sets dominate others.
   • They are useful for understanding the boundaries of set interactions.

8. Absorption Laws

   • A set unioned with the intersection of itself and another set equals the original set: A ∪ (A ∩ B) = A, and a set intersected with the union of itself and another set equals the original set: A ∩ (A ∪ B) = A.
   • These laws help simplify complex expressions by absorbing redundant components.
   • They highlight the efficiency of set operations in reducing expressions.

9. De Morgan's Laws

   • The complement of a union is the intersection of the complements: (A ∪ B)' = A' ∩ B', and the complement of an intersection is the union of the complements: (A ∩ B)' = A' ∪ B'.
   • These laws provide a powerful tool for transforming expressions involving complements.
   • They are essential for understanding the duality in set operations and their implications.

10. Double Complement Law

    • The complement of the complement of a set returns the original set: (A')' = A.
    • This law emphasizes the idea of negation and its reversibility in set theory.
    • It is fundamental for understanding how complements function within the broader context of set identities.