Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof (resit) / 1 / b

Use Karnaugh maps with rows $wx$ and columns $yz$, both in Gray code order $00,01,11,10$. Juxtaposition means Boolean conjunction, $+$ means Boolean disjunction, and a bar means Boolean negation. The minimal sum of products forms areandFor $c$, the groups are $\{4,5,6,7\}$, $\{1,3,5,7\}$ and $\{0,1,8,9\}$. For $d$, take $\{8,9\}$, $\{2,3\}$, $\{2,6\}$, $\{0,2\}$ and the isolated minterm $\{5\}$. Every group contains only required ones. The last $c$ group wraps between the top and bottom map rows.

These covers are minimal: $c$ needs its three essential groups, forced by inputs 6, 3 and 8. For $d$, input 5 is isolated, while 9, 3 and 6 each force their respective pair; input 0 still needs a further group. These five groups also attain the minimum literal count of 16.