1. 1 3- De-Morgan’s Theorems 1.The complement of a product of variables is equal to the sum of the complements of the variables. 2. The complement of a sum of variables is equal to the product of the complement of the variables.

2. 2 3- Applying De-Morgan's Theorems The following procedure illustrates the application of DeMorgan’s theorems and Boolean algebra to the specific expression Step 1.

3. 3 3- Applying De-Morgan's Theorems

4. 4 EXAMPLE :

5. 5 4- Simplification using Boolean Algebra A simplified Boolean expression uses the fewest gates possible to implement a given expression. Example : using Boolean algebra techniques, simplify this expression: AB + A(B+C) + B(B+C) Solution : Step 1: apply the distributive law to the second & third term in the expression AB + AB+AC + BB+BC Step 2: Apply rule 7 (BB =B) to the fourth term. AB + AB+AC + B+BC Step 3: apply rule 5 (AB+AB= AB) to the first two terms AB+AC + B+BC Step 4: apply rule 10 (B+BC=B) to the last two terms:- AB+AC+B Step 5 : apply rule 10 (AB+B=B) to the first and third terms:- B+AC

6. 6 4- Simplification using Boolean Algebra Implementation of the Boolean expression : AB + A(B+C) + B(B+C) = B+AC

7. 7 4- Simplification using Boolean Algebra

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10. 10 5- Standard forms of Boolean expression 5-1 The sum of product form (SOP) : When two or more product terms are summed by Boolean addition. The resulting expression is a sum-of-products (SOP). For example: ABC + CDE + B’CD’ Conversion of a general expression to SOP form : Any logic expression can be changed to SOP form by applying Boolean algebra techniques. For example the expression A(B +CD) can be converted to SOP form by applying the distributive law : A(B+CD) = AB + ACD Step 1: Multiply each nonstandard product term by a term made up of the sum of a missing variable and its complement. This results in two product terms. Step 2: Repeat step -1- until all resulting product term contain all variables in the domain in either complemented or un-complemented form.

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12. 12 5- Standard forms of Boolean expression 5-2 The Product of Sums (POS) form : When two or more sum terms are multiplied. The resulting expression is a Product Of Sums (POS). Examples : Converting a sum term to standard POS : Step 1 : Add to each nonstandard product term a term made up of the product of the missing variable and its complement. Step 2 : Apply rule 12 : A+BC = (A+B)(A+C). Step 3 : Repeat step -1- until all resulting sum terms contains all variables in the domain in either complemented or un-complemented form.

13. 13 5- Standard forms of Boolean expression Converting a sum term to standard POS

14. 14 5- Standard forms of Boolean expression 5-3 Converting Standard SOP to Standard POS: Step 1 :Evaluate each product sum in the SOP expression. That is determine the binary numbers that represent the product form. Step 2 : Determine all of the binary numbers not included in the evaluation in step 1. Step 3 : Write the equivalent sum term for each binary numbers from step 2 and express in pos form.

15. 15 5-3 Converting Standard SOP to Standard POS (cont.) : 5- Standard forms of Boolean expression

16. 16 6- Determining standards form truth tables There are four 1 in the output column and the corresponding binary values are 011, 100, 110 and 111. These binary values are converted to product term as :

17. 17 6- Determining standards form truth tables For the POS expression, the output is 0 for binary values 000,001,010 and 101. These binary values are converted to sum terms as follows:

18. 18 7- Karnaugh Map The Karnaugh map provides a systematic method for simplifying Boolean expressions and if properly used, will produce the simplest SOP or POS expression possible, known as the minimum expression.

19. 19 Cell Adjacency : 5-Variable Karnaugh Mapping

20. 20 8- Karnaugh Map SOP Minimization Example :-Map the standard SOP expression on a Karnaugh map Solution :- The expression is evaluated and a 1 is placed on the 3 variable Karnaugh map.

21. 21 8- Karnaugh Map SOP Minimization Example :- Map the following not standard SOP. Solution:- The SOP is not a standard form because each product term does not have three variables.

22. 22 8-1 Karnaugh Map Simplification of SOP Grouping the 1’s : I.A group must contain 1,2,4,8 or 16 cells. II.Each cell in a 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. III.Always include the largest possible number of “1”s in a group in accordance with rule 1. IV.Each “1” on the map must be included in at least one group. The “1”s already in a group can be included in other group as long as the overlapping groups include no common “1”s.

23. 23 8-1 Karnaugh Map Simplification of SOP

24. 24 8-2 Determining the minimum SOP expression from the map 1.Group the cells that have “1”s. Each group of cells containing “1”s creates one product term composed of all variables that occur in only one form. 2.Determine the minimum product term for each group I.A 1-cell group yield’s a 3-varaible product term II.A 2-cell group yield's a 2-variable product term. III.A 4-cell group yield's a 1-variable product term. IV.An 8-cell group yield’s a value of 1 for the expression. 3. When all the minimum product terms are derived from the Karnaugh map, they are summed from the minimum SOP expression.

25. 25 Example :- Determine the product terms for the Karnaugh map and write the resulting SOP minimum exp. Solution :- 8-2 Determining the minimum SOP expression from the map

26. 26 8-2 Determining the minimum SOP expression from the map Example :- Use Karnaugh map to minimize the following standard SOP Expression. Solution : The binary values of the expression are 101 + 011 + 001 +000 +100 The result is :- Example:-

27. 27 8-3 Mapping directly from truth table

28. 28 9- Karnaugh Map POS Minimization Mapping a standard POS Expression :- For a POS expression in standard form a “0” is placed on the Karnaugh map for each sum term in the expression. The mapping process can be shown :- Step1 :- Determine the binary values of each sum term in the standard POS expression, this is the binary value that makes the term equal to “0”. Step2 :- As each sum term is evaluated. Place a “0” on the Karnaugh map in the corresponding cell.

29. 29 9- Karnaugh Map POS Minimization

30. 30 Example :- Use Karnaugh map to minimize the following POS expr. Solution :-The combinations of binary values of the expression are (0 + 0 + 0)(0 + 0 + 1)(0 +1 +0)(0 +1 +1)(1 +1 +0) 9- Karnaugh Map POS Minimization

31. 31 Example:- Using a Karnaugh map convert POS expression into a minimum POS, to SOP expression and a minimum SOP expression. Solution:-

32. 32 Don’t Care Conditions