To simplify the given Boolean expression:

$(A + C)(A + \overline{A}C) + AC + C$

we will proceed step by step and state the Boolean identity used at each step.

Step 1: Expand the first part $(A + C)(A + \overline{A}C)$

Using the Distributive Law:

$(A + C)(A + \overline{A}C) = A(A + \overline{A}C) + C(A + \overline{A}C)$

Step 2: Apply Absorption Law to simplify terms

$A(A + \overline{A}C) = A$ because $A + \overline{A}C = A$ (Absorption).

Similarly: $C(A + \overline{A}C) = C$ because $A + \overline{A}C = A + C$ simplifies to $1$ using Complement Law.

So: $(A + C)(A + \overline{A}C) = A + C$

Step 3: Substitute back into the original expression

Now the expression becomes: $A + C + AC + C$

Step 4: Combine like terms

Use the Idempotent Law to combine $C + C = C$, so: $A + C + AC$

Step 5: Apply Absorption Law again

Notice that $A + AC = A$. This follows from $A(1 + C) = A$ by Identity Law.

Thus, the final simplified expression is: $A + C$

Summary of Boolean Identities used:

1. Distributive Law: $(A + B)(C + D) = AC + AD + BC + BD$
2. Absorption Law: $A + AB = A$
3. Complement Law: $A + \overline{A} = 1$
4. Idempotent Law: $A + A = A$
5. Identity Law: $A1 = A$

Would you like more details or have any questions?

Related Questions:

1. How does the Distributive Law work in Boolean Algebra?
2. What is the significance of the Absorption Law in simplifications?
3. Can you explain the Complement Law with more examples?
4. Why does $A + AC = A$ according to the Absorption Law?
5. How do the Idempotent and Identity Laws simplify Boolean expressions?

Tip: When simplifying Boolean expressions, always look for opportunities to apply the Absorption Law, as it can significantly reduce the complexity of the expression.