SOP/POS forms and canonical form of A+BC+ABC

SOP and POS forms

• Sum of Products (SOP): a logical sum (OR) of product (AND) terms, e.g. $F = AB + \overline{A}C$. It is built as an AND-OR (or NAND-NAND) circuit. If every term has all the variables, it is the canonical SOP (sum of minterms).
• Product of Sums (POS): a logical product (AND) of sum (OR) terms, e.g. $F = (A+B)(\overline{A}+C)$. It is built as an OR-AND (or NOR-NOR) circuit. If every term has all the variables, it is the canonical POS (product of maxterms).

Canonical form of $F = A + BC + ABC$

Expand each term to contain A, B and C:

$$\begin{aligned} A &= A(B+\overline{B})(C+\overline{C}) = ABC + AB\overline{C} + A\overline{B}C + A\overline{B}\,\overline{C} \\ BC &= BC(A+\overline{A}) = ABC + \overline{A}BC \\ ABC &= ABC \end{aligned}$$

Add them and keep each term once ($X+X=X$):

$$F = \overline{A}BC + A\overline{B}\,\overline{C} + A\overline{B}C + AB\overline{C} + ABC$$

Canonical SOP:

$$F = \Sigma m(3,4,5,6,7)$$

Canonical POS: the missing numbers 0, 1, 2 are the maxterms:

$$F = \Pi M(0,1,2) = (A+B+C)(A+B+\overline{C})(A+\overline{B}+C)$$

(The given function itself simplifies to $F = A + BC$, since $ABC$ is absorbed by $A$.)