Let's analyze the circuit and determine the Boolean expression step by step:

1. AND Gates:

   • The top AND gate receives inputs $X_1$ and $X_2$. Its output will be $X_1 \cdot X_2$.
   • The bottom AND gate receives inputs $X_3$ and $X_4$. Its output will be $X_3 \cdot X_4$.

2. OR Gate:

   • The outputs of the two AND gates ($X_1 \cdot X_2$ and $X_3 \cdot X_4$) serve as inputs to the OR gate. The output of the OR gate will be: $(X_1 \cdot X_2) + (X_3 \cdot X_4)$

3. Final AND Gate:

   • The output of the OR gate is one input to the final AND gate, while the second input to this gate comes directly from $X_1$.
   • The output of the final AND gate is: $Y = X_1 \cdot \big((X_1 \cdot X_2) + (X_3 \cdot X_4)\big)$

4. Simplification (if needed):

   • Expanding the expression: $Y = (X_1 \cdot X_1 \cdot X_2) + (X_1 \cdot X_3 \cdot X_4)$
   • Simplify $X_1 \cdot X_1$ to $X_1$: $Y = (X_1 \cdot X_2) + (X_1 \cdot X_3 \cdot X_4)$

Thus, the final Boolean expression for $Y$ is: $Y = (X_1 \cdot X_2) + (X_1 \cdot X_3 \cdot X_4)$

Would you like a detailed explanation for any part of the derivation?

Related Questions:

1. How does an AND gate differ from an OR gate in Boolean logic?
2. What would the output expression be if an XOR gate replaced the OR gate?
3. Can this circuit's output be minimized further using Boolean algebra rules?
4. How do truth tables help verify a Boolean circuit's output?
5. How does the addition of a NOT gate change the circuit's functionality?

Tip:

Always check the connections carefully in circuits—mismatching input/output relationships can lead to errors in deriving Boolean expressions!