For the digital block shown in the figure the output $\mathrm{Y}=f\left(\mathrm{~S}_{3}, \mathrm{~S}_{2}, \mathrm{~S}_{1}, \mathrm{~S}_{0}\right)$ where $\mathrm{S}_{3}$ is MSB and $\mathrm{S}_{0}$ is LSB. $Y$ is given in terms of minterms as $Y=$ $\sum m(1,5,6,7,11,12,13,15)$ and its complement is $\overline{\mathrm{Y}}=\sum m(0,2,3,4,8,9,10,14)$.
- Enter the logical values in the given Karnaugh map in the figure, for the output $Y$.
- Write down the expression for $Y$ in sum-of-products from using minimum number of terms.
- Draw the circuit for the digital logic boxes using four $2$-input NAND gates only for each of the boxes.