Question

Consider 4-variable functions \(f_1, f_2, f_3, f_4\) expressed in sum-of-minterms form as given below. \[ f_1 = \Sigma(0,2,3,5,7,8,11,13)
f_2 = \Sigma(1,3,5,7,11,13,15)
f_3 = \Sigma(0,1,4,11)
f_4 = \Sigma(0,2,6,13) \] The circuit is represented as: \begin{center} \includegraphics[width=6cm]{50.png} \end{center} With respect to the circuit given above, which of the following options is/are CORRECT?

• A. \(Y = \Sigma(0,1,2,11,13)\)
• B. \(Y = \Pi(3,4,5,6,7,8,9,10,12,14,15)\)
• C. \(Y = \Sigma(0,1,2,3,4,5,6,7)\)
• D. \(Y = \Pi(8,9,10,11,12,13,14,15)\)

Hint:

When solving circuit problems involving logical operations, calculate intermediate results step by step and carefully apply XOR and other operations.

Answer 1

The Correct Option is C

Step 1: Calculate the result of \(f_1 \land f_2\): \[ f_1 \land f_2 = \Sigma(3,5,7,11,13). \]
Step 2: Calculate the result of \(f_3 \lor f_4\): \[ f_3 \lor f_4 = \Sigma(0,1,2,4,6,11,13). \]
Step 3: XOR the outputs of \(f_1 \land f_2\) and \(f_3 \lor f_4\): The XOR operation results in the minterms present in one function but not both: \[ Y = \Sigma(0,1,2,3,4,5,6,7). \]
Analysis of Options: Option (A): Incorrect, as \(Y\) is not equal to \(\Sigma(0,1,2,11,13)\). Option (B): Incorrect, as \(\Pi(3,4,5,6,7,8,9,10,12,14,15)\) does not match \(Y\). Option (C): Correct, as \(Y = \Sigma(0,1,2,3,4,5,6,7)\). Option (D): Correct, since \(Y = \Sigma(0,1,2,3,4,5,6,7)\) implies the complement is: \[ Y = \Pi(8,9,10,11,12,13,14,15). \] Final Answer: \[ \boxed{\text{(C), (D)}} \]