Let's simplify the Boolean algebra expression step by step:

Expression:

$A \cdot A + \overline{A} \cdot \overline{A} + A \cdot B + \overline{A} \cdot C$

1. Simplify $A \cdot A$ and $\overline{A} \cdot \overline{A}$:

   • $A \cdot A = A$ (Idempotent Law)
   • $\overline{A} \cdot \overline{A} = \overline{A}$ (Idempotent Law)

   So the expression becomes: $A + \overline{A} + A \cdot B + \overline{A} \cdot C$

2. Simplify $A + \overline{A}$:

   • $A + \overline{A} = 1$ (Complement Law)

   So the expression reduces to: $1 + A \cdot B + \overline{A} \cdot C$

3. Simplify $1 + \text{anything}$:

   • $1 + A \cdot B + \overline{A} \cdot C = 1$ (Annulment Law)

Thus, the simplified Boolean expression is: $1$

This means the output of the expression is always true regardless of the values of $A$, $B$, and $C$.

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. What is the Idempotent Law in Boolean algebra?
2. How does the Complement Law work in Boolean algebra?
3. Can you explain the Annulment Law in Boolean algebra?
4. What would happen if we changed $B$ or $C$ in the original expression?
5. How do we apply De Morgan's laws in simplifying Boolean expressions?
6. What is the significance of the identity $A + \overline{A} = 1$?
7. What is a truth table, and how would it look for this expression?
8. How can Boolean algebra be applied in digital circuit design?

Tip: Always simplify Boolean expressions by starting with the basic laws like Idempotent, Complement, and Annulment. These help reduce the expression quickly.