Rangkuman 7.Aljabar Boolean, Penyederhanaan Logika dan Peta Karnaugh.

             Standard Forms of Boolean Expressions:

• 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

        AB + ABC

        AB + CDE + BCD

Conversion of a General Expression to SOP Form

Any logic expression can be change into SOP form by applying Boolean Algebra techniques

contoh : A(B + CD) = AB + ACD

Try this (A+B)+C=(A+B)C

                            =(A+B)C

                           =AC+BC

            

        The Standard SOP Form

           The Products-of-Sum (POS) Form

        When two or more sum terms are multiplied.

            The Standard POS Form

    

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.

The Karnaugh Map

• Provides a systematic method for simplifying Boolean expressions 
• Produces the simplest SOP or POS expression 
• 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

• A 1 is placed on the K- Map for each product term in the expression.
• Each 1 is placed in a cell corresponding to the value of a product term 

K-Map Simplification of SOP Expressions:

• A group must contain either 1, 2, 4, 8 or 16 cells. 
• 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 
• Always include the largest possible number 1s in a group in accordance with rule 1 
• 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

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. 

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