Q.Example 2: Simplify the Boolean expression $Y = \overline{A}\,\overline{B}\,\overline{C} + \overline{A}\,\overline{B}C + \overline{A}B\overline{C} + A\overline{B}\,\overline{C} + AB\overline{C}$ and then draw the logic diagram using only NAND gates.

On the three-variable map the two $\overline{C}$ columns roll together into a quad ($\overline{C}$) and an overlapping pair in the $\overline{A}$ row gives $\overline{A}\,\overline{B}$; ORing them and applying the double-inversion trick lets the …

[!TLDR]

Group a rolling quad ($\overline{C}$) and an overlapping pair ($\overline{A}\,\overline{B}$); the SOP $Y = \overline{C} + \overline{A}\,\overline{B}$ is then realised with NAND gates only.

Plotting the map

Grouping

• Rolling quad: the first column $\overline{B}\,\overline{C}$ and the last column $B\overline{C}$ are adjacent by rolling, so their four 1s form a quad. $A$ and $B$ change; the constant $C=0$ gives $\overline{C}$.
• Overlapping pair: in the $\overline{A}$ row the 1s in the $\overline{B}\,\overline{C}$ and $\overline{B}C$ columns form a pair. $C$ changes; constants $A=0,\ B=0$ give $\overline{A}\,\overline{B}$. It reuses (overlaps) the $(\overline{A},\ \overline{B}\,\overline{C})$ cell already in the quad.

ORing the two groups gives $Y = \overline{C} + \overline{A}\,\overline{B}$.

Realising with NAND gates only (Figure 10.4.2)

An AND-OR expression is converted to all-NAND logic by inverting each variable with a NAND and summing the product terms with a NAND (double inversion leaves the value unchanged): …