#### Answer

If we use the circuit construction algorithm described in this chapter, we produce the circuit: $\overline{a} \cdot \overline{b}+\overline{a} \cdot b .$
When this expression is implemented as a logic circuit, it takes two $NOT$ gates, two $AND$ gates, and one $OR$ gate for a total of $five$ gates.
However, looking carefully at the truth table, we see that the output is a $1$ whenever $a$ is a $0$, and the output is a $0$ whenever $a$ is a $1$.
The output is not affected by the value of $ b$. Thus, an equivalent circuit is $\overline{a},$ which takes only 1 gate-an improvement of $80 \%$.
This is a good example of how much optimization can improve a preliminary
design, and how important it can be to the efficiency of a computer system.