Let's analyze the circuit and determine the most simplified form of the output $D$. I will break this down step-by-step:

1. Logic Gates Observed

The circuit consists of:

• AND gates
• OR gates
• Connections of $A$, $B$, and $C$.

2. Step-by-Step Simplification

Gate 1 (Top-left AND gate)

This gate takes $A$ and $B$ as inputs: $\text{Output} = AB$

Gate 2 (Top-right OR gate)

This gate takes $AB$ (from Gate 1) and $B$ as inputs: $\text{Output} = AB + B$ Simplifying using the distributive property ($AB + B = B$): $\text{Output} = B$

Gate 3 (Bottom-left AND gate)

This gate takes $A$ and $C$ as inputs: $\text{Output} = AC$

Gate 4 (Final OR gate)

This OR gate takes $AC$ (from Gate 3) and $B$ (from Gate 2) as inputs: $\text{Output} = AC + B$

3. Final Output for $D$

The most simplified form of $D$ is: $D = AC + B$

4. Answer

From the options provided:

• Correct Option: (A) $B(A + C)$

This matches $AC + B$ when factored.

Would you like further clarification or additional breakdowns? Let me know!

Here are some related questions for practice:

1. How can the distributive property simplify Boolean expressions?
2. What is the difference between the output of an AND gate and an OR gate?
3. Why does $AB + B$ simplify to $B$?
4. How does factoring work in Boolean algebra (e.g., $B(A + C)$)?
5. How would you construct the truth table for this circuit?

Tip: Practice simplifying Boolean expressions using truth tables to verify your solutions!