Input bits X and Y are added by using the combinational logic as shown below. S represents the sum of the two bits. For a correct implementation of the sum, the signals \(D_0, D_1, D_2, D_3\) are _________, respectively.
Show Hint
- 1, 0, 0, 1
- 0, 1, 0, 1
- 1, 0, 1, 1
- 0, 1, 1, 0
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Circuit
The given logic diagram shows four AND gates whose outputs are ORed to produce the final sum \( S \). Each AND gate is enabled by one of the signals \( D_0, D_1, D_2, D_3 \). The gates are arranged such that each AND implements one minterm of the XOR truth table.
Step 2: Recall XOR Truth Table
The sum bit for adding two bits is \( S = X \oplus Y \). The truth table for XOR is:
| X | Y | S |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Thus, only minterms \( m_1 \) and \( m_2 \) must be active.
Step 3: Matching Minterms with \( D_0, D_1, D_2, D_3 \)
- \( D_0 \) corresponds to \( X' = 1, Y' = 1 \) → must be 0 because the XOR output is 0.
- \( D_1 \) corresponds to \( X' = 1, Y = 1 \) → must be 1 because the XOR output is 1.
- \( D_2 \) corresponds to \( X = 1, Y' = 1 \) → must be 1.
- \( D_3 \) corresponds to \( X = 1, Y = 1 \) → must be 0 because the XOR output is 0.
Step 4: Vector Check
Comparing with the given options, the correct configuration is:
\( D_0 = 1, D_1 = 0, D_2 = 0, D_3 = 1 \) which matches option (A).
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