GATE CS 2016 Set 1 Q30 — Combinational Circuits

To find the minimal sum of products (SOP) form of the output $X$, we analyze the cascaded $2$-to-$1$ multiplexers stage by stage.

Step 1: Analyze the First Multiplexer (Left)

The first multiplexer has:

• Select line: $P$
• Inputs: $I_0 = 0$ and $I_1 = R$

The output of a $2$-to-$1$ multiplexer with select line $S$ and inputs $I_0, I_1$ is given by the characteristic equation:

Substituting the values for the first multiplexer:

Step 2: Analyze the Second Multiplexer (Right)

The second multiplexer has:

• Select line: $Q$
• Inputs: $I_0 = \overline{R}$ and $I_1 = Y = PR$

Using the characteristic equation for the second multiplexer, the output $X$ is:

Step 3: Minimization

The expression $X = \overline{Q}\overline{R} + PQR$ cannot be simplified further because there are no common factors or consensus terms that can reduce the literal count. Thus, it is already in its minimal Sum of Products (SOP) form.

Therefore, the correct option is D.