Question

Complete the Truth Table below for the following
𝒀 = 𝑨 · (𝑩 + 𝑪
̅
)
A B C Y

Answer 1

To create the truth table for the expression \( Y = A \cdot (B + \overline{C}) \), we will evaluate \( Y \) for all possible values of \( A \), \( B \), and \( C \).

1. Determine \( \overline{C} \) (NOT C).
2. Determine \( B + \overline{C} \) (B OR NOT C).
3. Finally, calculate \( Y \) using \( A \) AND (B + NOT C).

Here is the complete truth table:

| A | B | C | \( \overline{C} \) | \( B + \overline{C} \) | Y = \( A \cdot (B + \overline{C}) \) |
|---|---|---|---------------------|-------------------------|--------------------------------------|
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 1 | 1 |

### Explanation of the Columns:
- **A**, **B**, **C**: These represent the possible values for the variables \( A \), \( B \), and \( C \) (0 for False, 1 for True).
- **\( \overline{C} \)**: The negation of \( C \).
- **\( B + \overline{C} \)**: The result of the OR operation between \( B \) and \( \overline{C} \).
- **Y**: The final output of the function based on the values of \( A \) and \( B + \overline{C} \).

Each row shows the output \( Y \) given specific inputs \( A, B, \) and \( C \).