Sum of Product (SOP)
Product of Sum (POS)
The Sum-of-Products (SOP) Form
When two or more product terms are summed by Boolean addition.
Conversion of a General Expression to SOP Form
Any logic expression be change into SOP form by applying Boolean Algebra
The Standard SOP Form
The Products-of-Sum (POS) Form
When two or more sum terms are multiplied.
Boolean Expression and Truth Table
Converting SOP to Truth Table
- Examine each of the products to determine where the product is equal to a 1.
- Set the remaining row outputs to 0.
Converting POS to Truth Table
- Opposite process from the SOP expressions.
- Each sum term results in a 0.
- Set the remaining row outputs to 1.
Converting from Truth Table to SOP and POS
The Karnaugh Map
l Provides a systematic method for simplifying Boolean expressions
l Produces the simplest SOP or POS expression
l Similar to a truth table because it presents all of the possible values of input variables
The 3-Variable K-Map
The 4-Variable K-Map
K-Map SOP Minimization
l A 1 is placed on the K- Map for each product term in the expression.
l Each 1 is placed in a cell corresponding to the value of a product term.
EXAMPLE:
Map the following standard SOP expression on a K-Map:
EXERCISE:
Map the following standard SOP expression on a K-Map
K-Map Simplification of SOP Expressions
l A group must contain either 1, 2, 4, 8 or 16 cells.
l Each cell in group must be adjacent to one or more cells in that same group but all cells in the group do not have to be adjacent to each other
l Always include the largest possible number 1s in a group in accordance with rule 1
l Each 1 on the map must be included in at least one group. The 1s already in a group can be included in another group as long as the overlapping groups include noncommon 1s
Determining the minimum SOP Expression from the Map
l Groups the cells that have 1s. Each group of cells containing 1s create one product term composed of all variables that occur in only one form (either uncomplemented or complemented) within the group. Variable that occurs both uncomplemented and complemented within the group are eliminated. These are called contradictory variables.
Example: Determine the product term for the K- Map below and write the resulting minimum SOP expression
Example: Use a K-Map to minimize the following standard SOP expression
Don’t Care (X) Conditions
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