Practice CircuitDrawing 1 .docx - Practice: Circuit Drawing Part 1 ... - Free Printable

Problem Analysis:

The task involves analyzing a logic circuit and answering questions related to its behavior. The circuit consists of two AND gates, one OR gate, and an inverter (NOT gate). The inputs are labeled as \( A \), \( B \), and \( C \), and the output is labeled as \( Y \).

#### Circuit Description:
1. First AND Gate: Inputs are \( A \) and \( B \).
2. Second AND Gate: Inputs are \( B \) and \( C \).
3. Inverter (NOT Gate): Input is \( C \).
4. OR Gate: Inputs are the outputs of the two AND gates and the inverted output of \( C \).
5. Final Output: \( Y \).

The overall expression for the output \( Y \) can be derived as follows:
- Output of the first AND gate: \( A \cdot B \)
- Output of the second AND gate: \( B \cdot C \)
- Output of the inverter: \( \overline{C} \)
- Output of the OR gate: \( (A \cdot B) + (B \cdot C) + \overline{C} \)

Thus, the final output \( Y \) is:
\[
Y = (A \cdot B) + (B \cdot C) + \overline{C}
\]

Task Breakdown:

We need to answer the following questions:

1. What are the values of \( A \), \( B \), and \( C \) when the output \( Y \) is 0?
2. Can we simplify the Boolean expression for \( Y \)? If so, how?
3. How many paths lead to \( Y = 1 \)?
4. If \( A = 1 \), what are the possible values of \( B \) and \( C \) such that \( Y = 1 \)?

---

Solution:

#### 1. What are the values of \( A \), \( B \), and \( C \) when the output \( Y \) is 0?

The output \( Y \) is given by:
\[
Y = (A \cdot B) + (B \cdot C) + \overline{C}
\]
For \( Y \) to be 0, all terms in the sum must be 0:
\[
(A \cdot B) = 0, \quad (B \cdot C) = 0, \quad \overline{C} = 0
\]

- From \( \overline{C} = 0 \), we get \( C = 1 \).
- With \( C = 1 \):
- \( (B \cdot C) = B \cdot 1 = B \). For \( B \cdot C = 0 \), we must have \( B = 0 \).
- \( (A \cdot B) = A \cdot 0 = 0 \). This is automatically satisfied regardless of \( A \).

Thus, the values are:
\[
A = \text{any value (0 or 1)}, \quad B = 0, \quad C = 1
\]

#### 2. Can we simplify the Boolean expression for \( Y \)? If so, how?

The expression for \( Y \) is:
\[
Y = (A \cdot B) + (B \cdot C) + \overline{C}
\]

Using the distributive property and simplification rules:
\[
Y = (A \cdot B) + (B \cdot C) + \overline{C}
\]
Factor out \( B \) from the first two terms:
\[
Y = B \cdot (A + C) + \overline{C}
\]

This is the simplified form of the expression.

#### 3. How many paths lead to \( Y = 1 \)?

To determine the number of paths leading to \( Y = 1 \), we analyze the conditions under which each term in the expression contributes to \( Y = 1 \):
\[
Y = (A \cdot B) + (B \cdot C) + \overline{C}
\]

- Path 1: \( A \cdot B = 1 \)
- This occurs when \( A = 1 \) and \( B = 1 \). The value of \( C \) does not matter.

- Path 2: \( B \cdot C = 1 \)
- This occurs when \( B = 1 \) and \( C = 1 \). The value of \( A \) does not matter.

- Path 3: \( \overline{C} = 1 \)
- This occurs when \( C = 0 \). The values of \( A \) and \( B \) do not matter.

Thus, there are three distinct paths leading to \( Y = 1 \).

#### 4. If \( A = 1 \), what are the possible values of \( B \) and \( C \) such that \( Y = 1 \)?

Given \( A = 1 \), the expression for \( Y \) becomes:
\[
Y = (1 \cdot B) + (B \cdot C) + \overline{C} = B + (B \cdot C) + \overline{C}
\]

Simplify:
\[
Y = B + \overline{C}
\]

For \( Y = 1 \):
- \( B = 1 \) (since \( B + \overline{C} \geq B \))
- Alternatively, \( \overline{C} = 1 \) (i.e., \( C = 0 \)).

Thus, the possible values are:
- \( B = 1 \) and \( C \) can be either 0 or 1.
- \( B = 0 \) and \( C = 0 \).

Summarizing:
\[
(B = 1, C = 0) \quad \text{or} \quad (B = 1, C = 1) \quad \text{or} \quad (B = 0, C = 0)
\]

---

Final Answers:

1. \( A = \text{any value}, B = 0, C = 1 \)
2. Simplified expression: \( Y = B \cdot (A + C) + \overline{C} \)
3. Number of paths: \( 3 \)
4. Possible values: \( (B = 1, C = 0), (B = 1, C = 1), (B = 0, C = 0) \)

\[
\boxed{
1. A = \text{any value}, B = 0, C = 1 \\
2. Y = B \cdot (A + C) + \overline{C} \\
3. 3 \\
4. (B = 1, C = 0), (B = 1, C = 1), (B = 0, C = 0)
}
\]