GATE CS 2021 Set 2 Q5 — Combinational Circuits

To implement the Boolean function $f(x,y,z) = \sum m(0,1,3,4,5,6)$ using a $4 \times 1$ multiplexer, we can choose $x$ and $y$ as the select lines ($S_1 = x$ and $S_0 = y$).

We determine the multiplexer inputs ($I_0, I_1, I_2, I_3$) based on the value of $z$ for each combination of $x$ and $y$:

• For $xy = 00$ ($I_0$): Covers minterms $m_0$ ($z=0$) and $m_1$ ($z=1$). Since both $m_0$ and $m_1$ are present in $f$, $I_0 = 1$.
• For $xy = 01$ ($I_1$): Covers minterms $m_2$ ($z=0$) and $m_3$ ($z=1$). Since only $m_3$ is present, $I_1 = z$.
• For $xy = 10$ ($I_2$): Covers minterms $m_4$ ($z=0$) and $m_5$ ($z=1$). Since both $m_4$ and $m_5$ are present, $I_2 = 1$.
• For $xy = 11$ ($I_3$): Covers minterms $m_6$ ($z=0$) and $m_7$ ($z=1$). Since only $m_6$ is present, $I_3 = \bar{z}$.

Thus, the multiplexer inputs are $I_0 = 1$, $I_1 = z$, $I_2 = 1$, and $I_3 = \bar{z}$ with select lines $S_1 = x, S_0 = y$. This configuration corresponds to the circuit shown in Option A.