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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.

Using the Identities and Laws of Boolean Algebra, simplify the following expression. List specific Boolean identity used for every simplification: (A + C)(A + AC) + AC + C

Explore the step-by-step simplification of the Boolean expression (A + C)(A + AC) + AC + C using fundamental laws like Distributive, Absorption, Complement, Idempotent, and Identity Laws of Boolean Algebra. This detailed guide is suitable for college students studying computer science or electrical engineering.

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