K-map Σm(0,1,2,8,10,14,15)+d, POS with NOR

Given $F(A,B,C,D) = \Sigma m(0,1,2,8,10,14,15) + d(3,7,11,13)$. The 0-cells are 4, 5, 6, 9, 12.

K-map (grouping the 0s for POS)

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

• Quad of 0s M(4,5,6,7) → $(A+\overline{B})$
• Quad of 0s M(4,5,12,13) → $(\overline{B}+C)$
• Pair of 0s M(9,11) → $(\overline{A}+B+\overline{D})$

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

Check: 4, 5, 6 by $(A+\overline{B})$; 12 by $(\overline{B}+C)$; 9 by $(\overline{A}+B+\overline{D})$. ($(\overline{A}+C+\overline{D})$, cells 9 and 13, is an equally good third term.)

Standard product of sums

The given 0s are $\Pi M(4,5,6,9,12)$ with don't-cares $d(3,7,11,13)$. The simplified POS also makes don't-cares 7, 11, 13 equal to 0, so its standard (canonical) POS is:

$$F = \Pi M(4,5,6,7,9,11,12,13)$$ $$\begin{aligned} F = {} & (A+\overline{B}+C+D)(A+\overline{B}+C+\overline{D})(A+\overline{B}+\overline{C}+D)(A+\overline{B}+\overline{C}+\overline{D}) \\ & (\overline{A}+B+C+\overline{D})(\overline{A}+B+\overline{C}+\overline{D})(\overline{A}+\overline{B}+C+D)(\overline{A}+\overline{B}+C+\overline{D}) \end{aligned}$$

Realization using NOR gates only

A POS expression maps directly to a two-level NOR-NOR circuit:

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

A ──[NOR]── A'   (inputs tied)
B ──[NOR]── B'   (inputs tied)
D ──[NOR]── D'   (inputs tied)

A, B'     ──[NOR]── P1  ─┐
B', C     ──[NOR]── P2  ─┤──[NOR]── F
A', B, D' ──[NOR]── P3  ─┘

Gates: 3 NOR inverters ($\overline{A}, \overline{B}, \overline{D}$), two 2-input NORs, one 3-input NOR and one 3-input output NOR: 7 NOR gates.