K-map Σ(2,3,6,7,9,10,11)+d in SOP and POS

Given $F(A,B,C,D) = \Sigma m(2,3,6,7,9,10,11) + d(5,8,12)$.

        CD
AB      00  01  11  10
      +---+---+---+---+
 00   | 0 | 0 | 1 | 1 |
      +---+---+---+---+
 01   | 0 | X | 1 | 1 |
      +---+---+---+---+
 11   | X | 0 | 0 | 0 |
      +---+---+---+---+
 10   | X | 1 | 1 | 1 |
      +---+---+---+---+

SOP form (grouping the 1s)

• Quad m(2,3,6,7) → $\overline{A}C$
• Quad m(8,9,10,11) → $A\overline{B}$

$$F = \overline{A}C + A\overline{B}$$

Don't-care 8 completes the $A\overline{B}$ quad; 5 and 12 are taken as 0.

POS form (grouping the 0s)

The 0-cells are 0, 1, 4, 13, 14, 15. Don't-cares 5 and 12 are used as 0 here (don't-care 8 is not grouped, so it stays 1):

• Quad of 0s M(0,1,4,5) → $(A+C)$
• Quad of 0s M(12,13,14,15) → $(\overline{A}+\overline{B})$

$$F = (A+C)(\overline{A}+\overline{B})$$

Check

Expanding the POS:

$$\begin{aligned} (A+C)(\overline{A}+\overline{B}) &= A\overline{A} + A\overline{B} + \overline{A}C + \overline{B}C \\ &= A\overline{B} + \overline{A}C + \overline{B}C \\ &= A\overline{B} + \overline{A}C \end{aligned}$$

($\overline{B}C$ is the consensus of $A\overline{B}$ and $\overline{A}C$, so it drops out.) Both forms describe exactly the same function: each sets don't-care 8 to 1 and don't-cares 5 and 12 to 0.

Each form needs only two 2-input gates plus one 2-input output gate (and inverters).