Question 13.11: Realizinga Product-of-Sums UsingOnly NAND Gates Realize the ...
Realizinga Product-of-Sums UsingOnly NAND Gates
Realize the following function in sum-of-products form, using only two-input NAND gates.
f=(\bar{x}+\bar{y}) \cdot(y+\bar{z})
Known quantities: f(x, y, z).
Find: Logic circuit for f using only NAND gates.
Analysis: The first step is to convert the expression for f into an expression that can be easily implemented with NAND gates. We observe that direct application of De Morgan’s theorem yields:
\begin{aligned}&\bar{x}+\bar{y}=\overline{x \cdot y} \\&y+\bar{z}=\overline{z \cdot \bar{y}}\end{aligned}
Thus, we can write the function as follows:
f=(\overline{x \cdot y}) \cdot(\overline{z \cdot \bar{y}})
and implement it with five NAND gates, as shown in Figure 13.41
Comments: Note that we used two NAND gates as inverters-one to obtain \bar{y}, the other to invert the output of the fourth NAND gate, equal to \overline{(\overline{x \cdot y}) \cdot(\overline{z \cdot \bar{y}})}.
