The Axiom of Unions Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proposition. Let C be a family of sets and let C ∈ C . Then � C ⊆ C . Proof. We must prove that every element of C is an element of � C , too. So let x ∈ C be arbitrary but fixed. By definition of � the union, x ∈ C . Because x ∈ C was arbitrary, every element � of C is in C . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proposition. Let C be a family of sets and let C ∈ C . Then � C ⊆ C . Proof. We must prove that every element of C is an element of � C , too. So let x ∈ C be arbitrary but fixed. By definition of � the union, x ∈ C . Because x ∈ C was arbitrary, every element � � C . Thus C ⊆ of C is in C . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proposition. Let C be a family of sets and let C ∈ C . Then � C ⊆ C . Proof. We must prove that every element of C is an element of � C , too. So let x ∈ C be arbitrary but fixed. By definition of � the union, x ∈ C . Because x ∈ C was arbitrary, every element � � C . Thus C ⊆ of C is in C . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. Let A , B , C be sets. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. Let A , B , C be sets. Then A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. Let A , B , C be sets. Then A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C. Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. Let A , B , C be sets. Then A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C. Proof. Simple exercise. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. Let A , B , C be sets. Then A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C. Proof. Simple exercise. (Maybe too simple.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Commutativity of set union. Let A , B be sets. Then A ∪ B = B ∪ A. Proof. Simple exercise. (Maybe too simple.) Theorem. Associativity of set union. Let A , B , C be sets. Then A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C. Proof. Simple exercise. (Maybe too simple.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Distributivity of unions over intersections and vice versa. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Distributivity of unions over intersections and vice versa. Let A , B , C be sets. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Distributivity of unions over intersections and vice versa. Let A , B , C be sets. Then the following hold. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Distributivity of unions over intersections and vice versa. Let A , B , C be sets. Then the following hold. 1. A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. Distributivity of unions over intersections and vice versa. Let A , B , C be sets. Then the following hold. 1. A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) . 2. A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B � � � � � � � � � � � A ∩ B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B A � � � � � � � � � � � ∪ A ∩ B C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B A � � � � � � � � � � � � � � � � � � � � � ∪ A ∩ B A ∩ C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B A A B � � � � � � � � � � � � � � � � � � � � � ∪ = A ∩ B A ∩ C C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B A A B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∪ = A ∩ B A ∩ C A ∩ ( B ∪ C ) C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � � A � � B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � A ∪ B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � � A � � B A � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∩ A ∪ B C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � � � � � A � � B A � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∩ A ∪ B A ∪ C � � C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � � � � � A � � B A � � A B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∩ = A ∪ B A ∪ C � � C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � � � � � � � � A � � B A � � A � � B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∩ = A ∪ B A ∪ C A ∪ ( B ∩ C ) � � C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). A ∩ ( B ∪ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∨ x ∈ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : ( x ∈ A ∧ x ∈ B ) ∨ ( x ∈ A ∧ x ∈ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : ( x ∈ A ∧ x ∈ B ) ∨ ( x ∈ A ∧ x ∈ C ) � � = x ∈ A : ( x ∈ A ∩ B ) ∨ ( x ∈ A ∩ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : ( x ∈ A ∧ x ∈ B ) ∨ ( x ∈ A ∧ x ∈ C ) � � = x ∈ A : ( x ∈ A ∩ B ) ∨ ( x ∈ A ∩ C ) � � = x ∈ A : x ∈ ( A ∩ B ) ∪ ( A ∩ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : ( x ∈ A ∧ x ∈ B ) ∨ ( x ∈ A ∧ x ∈ C ) � � = x ∈ A : ( x ∈ A ∩ B ) ∨ ( x ∈ A ∩ C ) � � = x ∈ A : x ∈ ( A ∩ B ) ∪ ( A ∩ C ) = ( A ∩ B ) ∪ ( A ∩ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A ∩ ( B ∪ C ) = x ∈ A : x ∈ A ∩ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : ( x ∈ A ∧ x ∈ B ) ∨ ( x ∈ A ∧ x ∈ C ) � � = x ∈ A : ( x ∈ A ∩ B ) ∨ ( x ∈ A ∩ C ) � � = x ∈ A : x ∈ ( A ∩ B ) ∪ ( A ∩ C ) = ( A ∩ B ) ∪ ( A ∩ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. DeMorgan’s Laws . Let A , B , C be sets. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. DeMorgan’s Laws . Let A , B , C be sets. Then the following hold. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. DeMorgan’s Laws . Let A , B , C be sets. Then the following hold. 1. A \ ( B ∪ C ) = ( A \ B ) ∩ ( A \ C ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Theorem. DeMorgan’s Laws . Let A , B , C be sets. Then the following hold. 1. A \ ( B ∪ C ) = ( A \ B ) ∩ ( A \ C ) . 2. A \ ( B ∩ C ) = ( A \ B ) ∪ ( A \ C ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � A � B � � � � � � � � � � � � � � A \ B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � A � B A � � � � � � � � � � � � � � ∩ A \ B C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � A � B A � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∩ A \ B A \ C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � A � B A � A B � � � � � � � � � � � � � � � � � � � � � � � � � � � � = ∩ A \ B A \ C C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � A � B A � A � B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � = ∩ A \ B A \ C A \ ( B ∪ C ) C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization A B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � A � B � � � � � � � � � � � � � � � � � A \ B logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � A � B A � � � � � � � � � � � � � � � � � ∪ A \ B C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � A � B A � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∪ A \ B A \ C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � A � B A � A B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � = ∪ A \ B A \ C C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Visualization � � � � � A � B A � A � B � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � = ∪ A \ B A \ C A \ ( B ∩ C ) C C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). A \ ( B ∪ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A \ ( B ∪ C ) = x ∈ A : x ∈ A \ ( B ∪ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A \ ( B ∪ C ) = x ∈ A : x ∈ A \ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∪ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A \ ( B ∪ C ) = x ∈ A : x ∈ A \ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∨ x ∈ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A \ ( B ∪ C ) = x ∈ A : x ∈ A \ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ) ∧¬ ( x ∈ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A \ ( B ∪ C ) = x ∈ A : x ∈ A \ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ) ∧¬ ( x ∈ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ) ∧ x ∈ A ∧¬ ( x ∈ C ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions

Unions Distributivity DeMorgan’s Laws Boolean Algebra Proof (part 1 only). � � A \ ( B ∪ C ) = x ∈ A : x ∈ A \ ( B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∪ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ∨ x ∈ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ) ∧¬ ( x ∈ C ) � � = x ∈ A : x ∈ A ∧¬ ( x ∈ B ) ∧ x ∈ A ∧¬ ( x ∈ C ) � � = x ∈ A : x ∈ A \ B ∧ x ∈ A \ C logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Axiom of Unions