Question

To obtain the Boolean function \( F(X, Y) = XY + \overline{X \), the inputs \( P, Q, R, S \) in the figure should be:}
\includegraphics[width=0.5\linewidth]{26.png}

• A. \( 1010 \)
  (B) \( 1110 \)
  (C) \( 0110 \)
  (D) \( 0001 \)
• B.
• C.
• D.

Hint:

A 4:1 multiplexer can be used to implement any 2-variable Boolean function by appropriately selecting the input combinations.

Answer 1

The Correct Option is B

Step 1: Logic implementation using a 4:1 multiplexer. The output of a 4:1 MUX is: \[ F = \overline{S_1}\overline{S_0}I_0 + \overline{S_1}S_0I_1 + S_1\overline{S_0}I_2 + S_1S_0I_3, \] where \( S_1 = X \) and \( S_0 = Y \). Step 2: Boolean function decomposition. The given function is: \[ F(X, Y) = XY + \overline{X}. \] Expanding it for all combinations of \( X \) and \( Y \): \[ F = \overline{X}P + X\overline{Y}Q + XYR + XYS. \] Comparing with the MUX equation, we assign: \[ P = 1, Q = 1, R = 1, S = 0. \] Hence, the correct option is (B).