Boolean Algebra and Boolean Rings

Definition 26   The collection of all subsets of a universal set $U$ forms a Boolean algebra; that is, the following hold where $A'$ means $U\setminus A$

1. Associativity of both operations: $A\cup (B \cup C)=(A \cup B) \cup C$ and $A\cap (B \cap C)=(A \cap B) \cap C$
2. Commutativity of both operations: $A\cup B=B\cup A$ and $A\cap B=B\cap A$
3. Identity $\emptyset \cup A= A$ and $U\cap A = A$
4. Complements: $A\cup A'=U$ and $A\cap A'= \emptyset$
5. Distributivity: $A \cup (B\cap C) = (A\cup B) \cap (A \cup C)$ and $A\cap (B \cup C)= (A\cap B) \cup (A\cap C)$
6. DeMorgan Laws: $(A\cup B)'=A'\cap B'$ and $(A \cap B)'= A' \cup B'$.
7. Double complement: $(A')'=A$

Problem 40   Verify the Distributive law for $\cup$ over $\cap$ using an elementwise argument.

(2 Points )

Problem 41   Verify the DeMorgan law for $\cap$ using an elementwise argument.

(2 Points )

Problem 42   Verify the Double Complement law using an elementwise argument.

(2 Points )

Definition 27   The set of all subsets of $U$ forms a Boolean ring under the operations $\bigtriangleup$ and $\cap$: in addition to the Associative, commutative, and identity axioms for $\cap$ from the previous definition we have

1. Nilpotence: $A\bigtriangleup A=\emptyset$
2. Distributivity: $A\cap (B\bigtriangleup C) = (A\cap B) \bigtriangleup (A\cap C)$
3. Associativity of $\bigtriangleup$: $A\bigtriangleup (B \bigtriangleup C) = (A\bigtriangleup B) \bigtriangleup C$
4. Commutativity of $\bigtriangleup$: $A\bigtriangleup B= B \bigtriangleup A$
5. Identity for $\bigtriangleup$: $A \bigtriangleup \emptyset = A$

Problem 43   Verify the Distributive law for $\cap$ over $\bigtriangleup$ using the axioms for a Boolean Algebra and the definition

$$A \bigtriangleup B= (A\cap B')\cup(A'\cap B).$$

(4 Points )

Problem 44   Verify the Associative law for $\bigtriangleup$ using the axioms for a Boolean Algebra.

(4 Points )

Problem 45   Verify the Distributive law for $\cap$ over $\cup$ using the axioms for a Boolean Ring and the definition

$$A\cup B = A\bigtriangleup(B\bigtriangleup (A\cap B))$$

(4 Points )