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 / 1 / b / Solution

For data inputs $d_0,\dots ,d_7$ and selection bits $s_2{s}_1{s}_0$, use a three-to-eight binary decoder. Its output $e_i$ is the minterm that is one exactly when the selection equals the three-bit binary representation of $i$. For example,Gate each data input with its decoded selection using an AND gate, then combine all eight results with an OR gate:Exactly one $e_i$ is asserted, so $z=d_i$ for the selected $i$, regardless of the other data inputs. A direct implementation uses three shared NOT gates, eight three-input AND gates for the binary decoder, eight two-input AND gates for the data, and one eight-input OR gate. Equivalently, combine each decoder term with its data input in one four-input AND gate. Larger logic gates may be built from two-input ones.