Canonical form of A(B'+C)+BD; minterm/maxterm

Canonical form of $F = A(\overline{B}+C) + BD$

Step 1: write F in SOP form

$$F = A\overline{B} + AC + BD$$

Step 2: expand each term to include all four variables A, B, C, D

$$\begin{aligned} A\overline{B} &= A\overline{B}(C+\overline{C})(D+\overline{D}) = A\overline{B}CD + A\overline{B}C\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}\,\overline{C}\,\overline{D} \\ &= m_{11}+m_{10}+m_9+m_8 \\ AC &= AC(B+\overline{B})(D+\overline{D}) = ABCD + ABC\overline{D} + A\overline{B}CD + A\overline{B}C\overline{D} \\ &= m_{15}+m_{14}+m_{11}+m_{10} \\ BD &= BD(A+\overline{A})(C+\overline{C}) = ABCD + AB\overline{C}D + \overline{A}BCD + \overline{A}B\overline{C}D \\ &= m_{15}+m_{13}+m_7+m_5 \end{aligned}$$

Step 3: combine, keeping each minterm once

Canonical SOP:

$$F = \Sigma m(5,7,8,9,10,11,13,14,15)$$ $$\begin{aligned} F = {} & \overline{A}B\overline{C}D + \overline{A}BCD + A\overline{B}\,\overline{C}\,\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}C\overline{D} \\ & + A\overline{B}CD + AB\overline{C}D + ABC\overline{D} + ABCD \end{aligned}$$

Canonical POS (the remaining numbers):

$$F = \Pi M(0,1,2,3,4,6,12)$$ $$\begin{aligned} F = {} & (A+B+C+D)(A+B+C+\overline{D})(A+B+\overline{C}+D)(A+B+\overline{C}+\overline{D}) \\ & (A+\overline{B}+C+D)(A+\overline{B}+\overline{C}+D)(\overline{A}+\overline{B}+C+D) \end{aligned}$$

Minterm and maxterm

• Minterm: a product term containing all variables once (true or complemented); it is 1 for exactly one input combination. Example: $m_5 = \overline{A}B\overline{C}D$ is 1 only for ABCD = 0101.
• Maxterm: a sum term containing all variables once (true or complemented); it is 0 for exactly one input combination. Example: $M_0 = A+B+C+D$ is 0 only for ABCD = 0000.
• $M_i = \overline{m_i}$; canonical SOP is a sum of minterms and canonical POS is a product of maxterms.