11. Standard Forms: SOP and POS

Before we start simplifying circuits graphically, we need to standardize the format of Boolean expressions. The two primary standard forms are Sum-of-Products (SOP) and Product-of-Sums (POS).

1. Sum-of-Products (SOP)

SOP is a series of AND terms (products) summed together (ORed).

• Example: $Y = A\overline{B} + \overline{A}C + B C D$
• Minterm: A product term where all variables appear exactly once, either complemented or uncomplemented. For 3 variables (A, B, C), $A\overline{B}C$ is a minterm.
• Canonical SOP: An expression composed entirely of minterms.

2. Product-of-Sums (POS)

POS is a series of OR terms (sums) multiplied together (ANDed).

• Example: $Y = (A + \overline{B}) \cdot (\overline{A} + C) \cdot (B + C + D)$
• Maxterm: A sum term where all variables appear exactly once. For 3 variables (A, B, C), $A + \overline{B} + C$ is a maxterm.
• Canonical POS: An expression composed entirely of maxterms.

Relationship between Minterms and Maxterms

If a function F is defined by a set of minterms ($\Sigma m$), its complement $\overline{F}$ is defined by the remaining minterms. Likewise, $F$ can be defined by the corresponding set of maxterms ($\Pi M$).

• If $F(A, B, C) = \Sigma(1, 4, 5)$, then $F(A, B, C) = \Pi(0, 2, 3, 6, 7)$.