https://file.aiquickdraw.com/mathbot/file/17264821228919hfdnmfg.pdf

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The document provided is a guide for simplifying logic circuits using Boolean algebra. It also includes tasks involving circuit creation and truth table construction. Let's work on simplifying the Boolean equations and constructing the truth tables.

### Task 1:
**Simplify and create a truth table for the Boolean expression:**
\[
X = (A + B + A' \cdot C) \cdot (A + C)
\]

#### Simplification:
1. Expand the expression using distribution:
   \[
   X = (A + B + A' \cdot C) \cdot (A + C)
   \]
   Applying distribution:
   \[
   X = (A \cdot A + A \cdot C) + (B \cdot A + B \cdot C) + (A' \cdot C \cdot A + A' \cdot C \cdot C)
   \]
2. Simplify terms using Boolean rules:
   - \(A \cdot A = A\)
   - \(A' \cdot C \cdot A = 0\) (since \(A \cdot A' = 0\))
   - \(A' \cdot C \cdot C = A' \cdot C\)
   
   The simplified equation becomes:
   \[
   X = A + A \cdot C + B \cdot A + B \cdot C + A' \cdot C
   \]
3. Applying absorption laws and further simplifications:
   - \(A + A \cdot C = A\) (absorption law)
   - \(B \cdot A + A = A\)
   
   So the final simplified expression is:
   \[
   X = A + B \cdot C + A' \cdot C
   \]

#### Truth Table:
We will now construct the truth table for the simplified expression \(X = A + B \cdot C + A' \cdot C\).

| A | B | C | B · C | A' | A' · C | X |
|---|---|---|-------|----|--------|---|
| 0 | 0 | 0 |   0   |  1 |    0   | 0 |
| 0 | 0 | 1 |   0   |  1 |    1   | 1 |
| 0 | 1 | 0 |   0   |  1 |    0   | 0 |
| 0 | 1 | 1 |   1   |  1 |    1   | 1 |
| 1 | 0 | 0 |   0   |  0 |    0   | 1 |
| 1 | 0 | 1 |   0   |  0 |    0   | 1 |
| 1 | 1 | 0 |   0   |  0 |    0   | 1 |
| 1 | 1 | 1 |   1   |  0 |    0   | 1 |

---

### Task 2:
**Simplify and create a truth table for the Boolean expression:**
\[
X = (A \cdot B + D) \cdot (C + A \cdot D)
\]

#### Simplification:
1. Expand the expression using distribution:
   \[
   X = (A \cdot B + D) \cdot (C + A \cdot D)
   \]
   Applying distribution:
   \[
   X = (A \cdot B \cdot C) + (A \cdot B \cdot A \cdot D) + (D \cdot C) + (D \cdot A \cdot D)
   \]
2. Simplify terms using Boolean rules:
   - \(A \cdot B \cdot A = A \cdot B\)
   - \(D \cdot A \cdot D = D \cdot A\)
   
   The simplified equation becomes:
   \[
   X = A \cdot B \cdot C + A \cdot B \cdot D + D \cdot C + D \cdot A
   \]
3. Applying absorption laws:
   - \(A \cdot B \cdot D + D \cdot A = D \cdot A\) (absorption)
   
   So the final simplified expression is:
   \[
   X = A \cdot B \cdot C + D \cdot C + D \cdot A
   \]

#### Truth Table:
We will now construct the truth table for the simplified expression \(X = A \cdot B \cdot C + D \cdot C + D \cdot A\).

| A | B | C | D | A · B · C | D · C | D · A | X |
|---|---|---|---|-----------|-------|-------|---|
| 0 | 0 | 0 | 0 |     0     |   0   |   0   | 0 |
| 0 | 0 | 0 | 1 |     0     |   0   |   0   | 0 |
| 0 | 0 | 1 | 0 |     0     |   0   |   0   | 0 |
| 0 | 0 | 1 | 1 |     0     |   1   |   0   | 1 |
| 0 | 1 | 0 | 0 |     0     |   0   |   0   | 0 |
| 0 | 1 | 0 | 1 |     0     |   0   |   0   | 0 |
| 0 | 1 | 1 | 0 |     0     |   0   |   0   | 0 |
| 0 | 1 | 1 | 1 |     0     |   1   |   0   | 1 |
| 1 | 0 | 0 | 0 |     0     |   0   |   0   | 0 |
| 1 | 0 | 0 | 1 |     0     |   0   |   1   | 1 |
| 1 | 0 | 1 | 0 |     0     |   0   |   0   | 

Learn how to simplify Boolean logic expressions and create truth tables using Boolean algebra rules. This guide walks through step-by-step simplifications for expressions such as X = (A + B + A' · C) · (A + C), demonstrating key theorems like De Morgan's Law and the absorption property. Ideal for electronics and digital logic students.

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