Minterms/maxterms; K-map Σm(0,1,2,5,8,14)+d, NOR

Minterms and maxterms

A minterm is a product (AND) term that contains every variable of the function exactly once, in true or complemented form. It equals 1 for exactly one input combination. A variable is written true if its value is 1 and complemented if it is 0; e.g. for ABC = 101 the minterm is $m_5 = A\overline{B}C$. A function in canonical SOP is a sum of minterms: $F = \Sigma m(\dots)$.

A maxterm is a sum (OR) term that contains every variable exactly once, in true or complemented form. It equals 0 for exactly one input combination. A variable is written true if its value is 0 and complemented if it is 1; e.g. for ABC = 101 the maxterm is $M_5 = \overline{A}+B+\overline{C}$. A function in canonical POS is a product of maxterms: $F = \Pi M(\dots)$.

$M_i = \overline{m_i}$, and a function's minterm list and maxterm list are complementary (together they hold all $2^n$ numbers).

K-map simplification

$F(A,B,C,D) = \Sigma m(0,1,2,5,8,14) + d(4,10,13)$

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

• Quad m(0,1,4,5) → $\overline{A}\,\overline{C}$
• Quad m(0,2,8,10) → $\overline{B}\,\overline{D}$
• Pair m(10,14) → $AC\overline{D}$

$$F = \overline{A}\,\overline{C} + \overline{B}\,\overline{D} + AC\overline{D}$$

Check: 0, 1, 5 by $\overline{A}\,\overline{C}$ (with 4); 2, 8 by $\overline{B}\,\overline{D}$ (with 10); 14 by $AC\overline{D}$ (with 10). Don't-care 13 is not used. (The minimal POS needs 4 terms, so the SOP is simpler.)

NOR-only implementation

For NOR-only realization of an SOP expression, write each product term as a NOR of complemented literals ($\overline{X}\,\overline{Y} = \overline{X+Y}$), feed these to a NOR gate (which gives $\overline{F}$), and invert with one more NOR whose inputs are tied.

$$\begin{aligned} \overline{A}\,\overline{C} &= \overline{A+C} \\ \overline{B}\,\overline{D} &= \overline{B+D} \\ AC\overline{D} &= \overline{\overline{A}+\overline{C}+D} \\ F &= \overline{\overline{P_1+P_2+P_3}} \end{aligned}$$

A ──[NOR]── A'   (inputs tied)
C ──[NOR]── C'   (inputs tied)

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

F' ──[NOR]── F   (inputs tied)

Gates: 2 NOR inverters, two 2-input NORs, one 3-input NOR, a 3-input NOR giving $\overline{F}$ and a NOR inverter: 7 NOR gates.