Step 1: Understanding the Question:
The task is to determine the logic output \(Y\) for a given combination of logic gates and inputs. Step 2: Key Formula or Approach:
Identify each gate in the circuit:
- Top part: Input \(A\) passes through a NOT gate, then is fed into an OR gate with input \(B\). Output \(X_1 = \bar{A} + B\).
- Bottom part: Inputs \(A\) and \(B\) are fed into an AND gate. Output \(X_2 = A \cdot B\).
- Final part: \(X_1\) and \(X_2\) are fed into a NAND gate. Output \(Y = \overline{X_1 \cdot X_2}\). Step 3: Detailed Explanation:
Boolean expression for the circuit:
\[ Y = \overline{(\bar{A} + B) \cdot (A \cdot B)} \]
By Boolean algebra laws:
\[ (\bar{A} + B) \cdot AB = \bar{A}AB + BAB = 0 + AB = AB \]
Thus, the circuit simplifies to \(Y = \overline{AB}\), which is a NAND operation on \(A\) and \(B\).
1. Case 1: \(A = 1, B = 1\):
\[ Y = \overline{1 \cdot 1} = \bar{1} = 0 \]
2. Case 2: \(A = 0, B = 1\):
\[ Y = \overline{0 \cdot 1} = \bar{0} = 1 \] Step 4: Final Answer:
The outputs are 0 and 1, respectively.
