Minterms/maxterms; K-map Σm(0,1,2,4,6,8,9,12,13,14)

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

$F(A,B,C,D) = \Sigma m(0,1,2,4,6,8,9,12,13,14)$

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

Grouping the 1s (SOP):

• Quad m(0,2,4,6) → $\overline{A}\,\overline{D}$
• Quad m(4,6,12,14) → $B\overline{D}$
• Quad m(8,9,12,13) → $A\overline{C}$
• Quad m(0,1,8,9) → $\overline{B}\,\overline{C}$

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

Check: 0, 2, 4, 6 by $\overline{A}\,\overline{D}$; 12, 14 by $B\overline{D}$; 8, 9, 12, 13 by $A\overline{C}$; 0, 1, 8, 9 by $\overline{B}\,\overline{C}$. All 10 minterms are covered by four quads.

Grouping the 0s (POS): the 0s are at 3, 5, 7, 10, 11, 15.

• Pair of 0s M(5,7) → $(A+\overline{B}+\overline{D})$
• Pair of 0s M(10,11) → $(\overline{A}+B+\overline{C})$
• Quad of 0s M(3,7,11,15) → $(\overline{C}+\overline{D})$

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

Both forms use 8 literals; either is an acceptable simplified answer.