Minterms/maxterms; K-map Σ(1,3,5,8,10,11,12)+d

Min-terms and max-terms

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

$F(A,B,C,D) = \Sigma m(1,3,5,8,10,11,12) + d(2,9,15)$

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

• Pair m(1,5) → $\overline{A}\,\overline{C}D$
• Pair m(8,12) → $A\overline{C}\,\overline{D}$
• Quad m(2,3,10,11) → $\overline{B}C$

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

Check: 3, 10, 11 by $\overline{B}C$ (with don't-care 2); 1, 5 by $\overline{A}\,\overline{C}D$; 8, 12 by $A\overline{C}\,\overline{D}$. Don't-cares 9 and 15 are not needed.

Logic circuit (AND-OR)

B' ─┐
C  ─┴─[AND]── B'C ────┐
A' ─┐                 │
C' ─┼─[AND]── A'C'D ──┼─[OR]── F
D  ─┘                 │
A  ─┐                 │
C' ─┼─[AND]── AC'D' ──┘
D' ─┘

Complements $\overline{A}, \overline{B}, \overline{C}, \overline{D}$ come from NOT gates. The circuit needs one 2-input AND, two 3-input ANDs and one 3-input OR (plus inverters).