K-map Σ(0,1,4,8,10,11,12)+d, standard minterm

Given $F(A,B,C,D) = \Sigma m(0,1,4,8,10,11,12) + d(2,3,6,9,15)$.

K-map simplification

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

Groups (don't-cares 2, 3, 9 complete the octet; 6 and 15 are not used):

• Octet m(0,1,2,3,8,9,10,11) → $\overline{B}$
• Quad m(0,4,8,12) → $\overline{C}\,\overline{D}$

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

Check: the octet covers 0, 1, 8, 10, 11 and the quad covers 0, 4, 8, 12.

Conversion of the result into standard minterm form

A standard (canonical) minterm form has every variable in every term. Multiply each term by $(X+\overline{X})$ for each missing variable X.

Term $\overline{B}$ (A, C, D missing):

$$\begin{aligned} \overline{B} &= \overline{B}(A+\overline{A})(C+\overline{C})(D+\overline{D}) \\ &= \overline{A}\,\overline{B}\,\overline{C}\,\overline{D} + \overline{A}\,\overline{B}\,\overline{C}D + \overline{A}\,\overline{B}C\overline{D} + \overline{A}\,\overline{B}CD \\ &\quad + A\overline{B}\,\overline{C}\,\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}C\overline{D} + A\overline{B}CD \\ &= m_0+m_1+m_2+m_3+m_8+m_9+m_{10}+m_{11} \end{aligned}$$

Term $\overline{C}\,\overline{D}$ (A, B missing):

$$\begin{aligned} \overline{C}\,\overline{D} &= (A+\overline{A})(B+\overline{B})\overline{C}\,\overline{D} \\ &= \overline{A}\,\overline{B}\,\overline{C}\,\overline{D} + \overline{A}B\overline{C}\,\overline{D} + A\overline{B}\,\overline{C}\,\overline{D} + AB\overline{C}\,\overline{D} \\ &= m_0+m_4+m_8+m_{12} \end{aligned}$$

Combine and drop repeated terms ($m_0$ and $m_8$ appear twice; $X+X=X$):

$$F = \Sigma m(0,1,2,3,4,8,9,10,11,12)$$ $$\begin{aligned} F = {} & \overline{A}\,\overline{B}\,\overline{C}\,\overline{D} + \overline{A}\,\overline{B}\,\overline{C}D + \overline{A}\,\overline{B}C\overline{D} + \overline{A}\,\overline{B}CD + \overline{A}B\overline{C}\,\overline{D} \\ & + A\overline{B}\,\overline{C}\,\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}C\overline{D} + A\overline{B}CD + AB\overline{C}\,\overline{D} \end{aligned}$$

This contains all 7 original minterms plus the don't-cares 2, 3, 9 that the simplification set to 1 (don't-cares 6 and 15 were taken as 0).