K-map Π(0,2,5,8,10)+d, SOP and NOR

Given $F(A,B,C,D) = \Pi M(0,2,5,8,10) + d(7,15)$. The listed maxterms are the 0s; the other cells (except don't-cares 7, 15) are 1s.

Standard SOP

The standard (canonical) SOP is the sum of the minterms not in the maxterm list:

$$F = \Sigma m(1,3,4,6,9,11,12,13,14) + d(7,15)$$ $$\begin{aligned} F = {} & \overline{A}\,\overline{B}\,\overline{C}D + \overline{A}\,\overline{B}CD + \overline{A}B\overline{C}\,\overline{D} + \overline{A}BC\overline{D} + A\overline{B}\,\overline{C}D \\ & + A\overline{B}CD + AB\overline{C}\,\overline{D} + AB\overline{C}D + ABC\overline{D} \end{aligned}$$

K-map

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

Grouping 1s (minimal SOP):

• Quad m(1,3,9,11) → $\overline{B}D$
• Quad m(4,6,12,14) → $B\overline{D}$
• Quad m(9,11,13,15) → $AD$

$$F = \overline{B}D + B\overline{D} + AD$$

(The last term may also be taken as $AB$, quad 12, 13, 14, 15; both are minimal.)

Grouping 0s (minimal POS):

• Quad of 0s M(0,2,8,10) → $(B+D)$
• Pair of 0s M(5,7) → $(A+\overline{B}+\overline{D})$

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

Implementation using NOR gates only

The POS form has fewer terms, and POS maps directly to a two-level NOR-NOR circuit:

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

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

B, D      ──[NOR]── P1  ─┐
                         ├──[NOR]── F
A, B', D' ──[NOR]── P2  ─┘

Gates: 2 NOR inverters, one 2-input NOR, one 3-input NOR and one 2-input output NOR: 5 NOR gates.

Check with $ABCD = 0101$ (m5): $B+D = 1$, $A+\overline{B}+\overline{D} = 0$, so F = 0, correct.