Rangkuman Konsep Digital: Tugas 7. Rangkuman aljaba Boolean, penyederhanaan logika dan peta karnaugh

Aljabar Boolean, Penyederhanaan Logika dan Peta Karnaugh

standart forms of Boolean expression

1.Sum of Product(SOP)

The Sum of Product(SOP) form when two or more product terms are summed by boolean addition.

conversion of a General Expression to SOP from

any logic expression can be change into SOP form by applying Boolean Algebra Techniques

example

A(A+CD)=AB+ACD

try this : (A+B)+C=(A+B)C

=(A+B)C

=AC + BC

The standart SOP form

2.Product of Sum(POS)

when two or more sum terms are multiplied.

The Standart POS Form

Boolean Expression and Truth Table

Converting SOP to Truth Table

1.Examine each of the products to determine where the product is equal to a 1.

2.Set the remaining row outputs to 0.

1.Opposite process from the SOP expressions.

2.Each sum term results in a 0.

3.Set the remaining row outputs to 1.

The Karnaugh Map

1.Provides a systematic method for simplifying Boolean expressions

2.Produces the simplest SOP or POS expression

3.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

1. A 1 is placed on the KMap for each product term in the expression.

2. 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

1. A group must contain either 1, 2, 4, 8 or 16 cells.

2.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

3. Always include the largest possible number 1s in a group in accordance with rule 1

4. 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.

Example: Group the 1s in each KMaps

Determining the minimum SOP

Expression from the Map

1.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 KMap below and write the resulting minimum SOP expression