### Problem 2: Analysis of the CircuitThe objective is to analyze the given logic circuit and complete the following tasks:#### Tasks:a) **Copy the image to Paint and label the signal at the output of every gate.**b) **Write the equivalent Boolean expression of the circuit.**c) **Write the dual of the Boolean expression of the circuit.**#### Circuit Diagram Breakdown:1. **Inputs:** - The circuit has two inputs labeled A and B.2. **Logic Gates:** - There are three NOT gates, one AND gate, and one OR gate in the circuit.3. **Gate Connections:** - The inputs A and B both pass through separate NOT gates. - The output of the A's NOT gate is fed into the first AND gate. - The output of the B's NOT gate also feeds into both the first AND gate and the OR gate. - The third NOT gate takes input directly from A and feeds its output into the OR gate. - The outputs of the first AND gate and the second NOT gate are inputs to a final OR gate, producing the circuit's final output.#### Outputs and Boolean Expressions:- For **task b**, we aim to write the Boolean expression: - Let: - $ \overline{A} $ be the NOT of A, - $ \overline{B} $ be the NOT of B. - From the circuit: - The output of the first AND gate: $ \overline{A} \cdot \overline{B} $ - The output of the OR gate: $ \overline{B} + \overline{A} $ - Thus, the overall Boolean expression: $ (\overline{A} \cdot \overline{B}) + \overline{B} + \overline{A} $- For **task c**, compute the dual of the Boolean expression by interchanging AND and OR operations, and replacing 0s with 1s and vice versa: - Original Expression: $ (\overline{A} \cdot \overline{B}) + \overline{B} + \overline{A} $ - Dual Expression: $ (\overline{A} + \overline{B}) \cdot \overline{B} \cdot \over **3. Consider the universal set:**\( U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} $Let set $ A = $ {all digits with a curve in the character}Let set $ B = $ {all digits with a straight line in the character}a) Draw a Venn diagram representing sets $ U, A, B $, with the digits in the appropriate areas of the diagram.b) List the elements of set $ F = B - A $.c) Evaluate $\sum x$ where $ x \in (A \cup B) - A \cap B $.

Answered: a) Copy the image to Paint and label…

a) Copy the image to Paint and label the signal at the output of every gate b) Write the equivalent Boolean expression of the circuit c) Write the dual of Boolean expression of the circuit

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Problem 2: Analysis of the Circuit

The objective is to analyze the given logic circuit and complete the following tasks:

#### Tasks:

a) **Copy the image to Paint and label the signal at the output of every gate.**

b) **Write the equivalent Boolean expression of the circuit.**

c) **Write the dual of the Boolean expression of the circuit.**

#### Circuit Diagram Breakdown:

1. **Inputs:**
- The circuit has two inputs labeled A and B.

2. **Logic Gates:**
- There are three NOT gates, one AND gate, and one OR gate in the circuit.

3. **Gate Connections:**
- The inputs A and B both pass through separate NOT gates.
- The output of the A's NOT gate is fed into the first AND gate.
- The output of the B's NOT gate also feeds into both the first AND gate and the OR gate.
- The third NOT gate takes input directly from A and feeds its output into the OR gate.
- The outputs of the first AND gate and the second NOT gate are inputs to a final OR gate, producing the circuit's final output.

#### Outputs and Boolean Expressions:

- For **task b**, we aim to write the Boolean expression:
- Let:
- $ \overline{A} $ be the NOT of A,
- $ \overline{B} $ be the NOT of B.
- From the circuit:
- The output of the first AND gate: $ \overline{A} \cdot \overline{B} $
- The output of the OR gate: $ \overline{B} + \overline{A} $
- Thus, the overall Boolean expression: $ (\overline{A} \cdot \overline{B}) + \overline{B} + \overline{A} $

- For **task c**, compute the dual of the Boolean expression by interchanging AND and OR operations, and replacing 0s with 1s and vice versa:
- Original Expression: $ (\overline{A} \cdot \overline{B}) + \overline{B} + \overline{A} $
- Dual Expression: \( (\overline{A} + \overline{B}) \cdot \overline{B} \cdot \over

**3. Consider the universal set:**

$ U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} $

Let set $ A = $ {all digits with a curve in the character}

Let set $ B = $ {all digits with a straight line in the character}

a) Draw a Venn diagram representing sets $ U, A, B $, with the digits in the appropriate areas of the diagram.

b) List the elements of set $ F = B - A $.

c) Evaluate $\sum x$ where $ x \in (A \cup B) - A \cap B $.

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