Answer
See explanation
Work Step by Step
We are asked to **prove** the identity:
\[
\boxed{a + b = \overline{\overline{a} \cdot \overline{b}}}
\quad \text{for all } a, b \in B
\]
This is a standard Boolean algebra identity:
### 🔷 De Morgan’s Law (Theorem 6.4.1 #6a), rewritten.
---
### ✅ Step-by-step Proof:
We want to prove:
\[
a + b = \overline{\overline{a} \cdot \overline{b}}
\]
We'll use:
- **De Morgan’s Law**:
\[
\overline{a + b} = \overline{a} \cdot \overline{b}
\Rightarrow
\text{Take complement of both sides: }
\boxed{a + b = \overline{\overline{a} \cdot \overline{b}}}
\]
This is valid because of the **Double Complement Law** (Theorem 6.4.1 #3):
\[
\overline{\overline{x}} = x
\]
---
### ✅ Final Answer:
\[
\boxed{a + b = \overline{\overline{a} \cdot \overline{b}}}
\quad \text{(by De Morgan’s Law and Double Complement Law)}
\]
