Lecture on Set Theory: Laws of Sets

Idempotent Laws

• Law 1: A Union A = A
  • Union of a set with itself results in the set itself.
  • Proof:
    • A union A = {x | x ∈ A or x ∈ A}
    • Repetition of statement, simplifies to {x | x ∈ A}, which is A.
• Law 2: A Intersection A = A
  • Intersection of a set with itself results in the set itself.
  • Proof:
    • A intersection A = {x | x ∈ A and x ∈ A}
    • Again, repetition, simplifies to {x | x ∈ A}, which is A.

Identity Laws

• Law 1: A Union Ø = A
  • Union with a null set results in the original set.
  • Proof:
    • A union Ø = {x | x ∈ A or x ∈ Ø}
    • Ø has no elements, so {x | x ∈ A}, which is A.
• Law 2: A Intersection U = A
  • Intersection with a universal set results in the original set.
  • Proof:
    • A intersection U = {x | x ∈ A and x ∈ U}
    • U contains all elements, hence {x | x ∈ A}, which is A.

Subset Property

• Two sets are equal if each is a subset of the other.

Commutative Laws

• Law 1: A Union B = B Union A
  • The order in union does not matter.
  • Proof by showing element-wise inclusion to and from A union B and B union A.
• Law 2: A Intersection B = B Intersection A
  • The order in intersection does not matter. (Self-practice)

Associative Laws

• Law 1: (A Union B) Union C = A Union (B Union C)
  • Grouping in union does not matter.
  • Proof uses arbitrary element x inclusion in both sides of the equation.
• Law 2: (A Intersection B) Intersection C = A Intersection (B Intersection C)
  • Grouping in intersection does not matter. (Self-practice)

Distributive Laws

• Law 1: A Union (B Intersection C) = (A Union B) Intersection (A Union C)
  • Union distributes over intersection.
  • Proof through arbitrary element x inclusion steps.
• Law 2: A Intersection (B Union C) = (A Intersection B) Union (A Intersection C)
  • Intersection distributes over union. (Self-practice)

De Morgan's Laws

• Law 1: (A Union B)' = A' Intersection B'
  • Complement of union is the intersection of complements.
  • Proof with element x not belonging to A union B and shown to belong to complement sets.
• Law 2: (A Intersection B)' = A' Union B'
  • Complement of intersection is the union of complements. (Self-practice)

Key Notes

• Complement Set: Contains all elements not in the original set.
• Universal Set: Contains all possible elements under consideration.
• Null Set (Ø): An empty set with no elements.

Exercise: Practice proving the unproven laws on your own for mastery.