Part#3_Logic_Design_2020.pdfPart#3_Logic_Design_2020.pdfPart#3_Logic_Design_2020.pdf
- 1. Boolean Algebra Prof. Dr. Khalid M. Hosny 1
- 2. 2 Variable – a symbol used to represent a logical quantity. Complement – the inverse of a variable and is indicated by a bar over the variable. Literal – a variable or the complement of a variable. Boolean Operations & Expressions
- 3. Basic Boolean Equations For the basic gates/functions AND ◦ Z = A B ◦ X = C D E 3 input gate ◦ Y = F G H K 4 input gate OR ◦ Z = A + B ◦ Y = F + G + H + K 4 input gate NOT F = 𝒁 3
- 4. Boolean Algebra theorems 1. X + 0 = X 2. X + 1 = 1 3. X · 0 = 0 4. X· 1 = X 5. X + X = X 6. X · X = X 7. X + 𝑿 = 1 8. X · 𝑿 = 0 9. 𝑿= X 4
- 5. Boolean Algebra theorems 10. X + Y = Y + X 11. X . Y = Y. X 12. X + Y + Z = X + Y + Z 13. X(YZ) = XY Z 14. X(Y + Z) = XY + XZ 15. X + XY = X 16. X + 𝑿Y = X + Y 17. X + YZ = (X + Y)(X + Z) 18. X + 𝑿Y = X 5
- 7. DeMorgan’s Theorems The complement of two or more ANDed variables is equivalent to the OR of the complements of the individual variables. The complement of two or more ORed variables is equivalent to the AND of the complements of the individual variables. Y X Y X Y X Y X NAND Negative-OR Negative-AND NOR 7
- 8. Very important in simplifying equations ◦ (𝑿 + 𝒀) = 𝑿· 𝒀 ◦ (𝑿𝒀)= 𝑿 + 𝒀 DeMorgan’s Theorems 8
- 9. Complement of a function In real implementation sometimes the complement of a function is needed. ◦ Have F=𝑿Y𝒁+𝑿𝒀Z 9
- 10. Apply DeMorgan’s theorems to the expressions: DeMorgan’s Theorems (Exercises) 10
- 11. DeMorgan’s Theorems (Exercises) Apply DeMorgan’s theorems to the expressions: ) ( ) ( F E D C B A EF D C B A DEF ABC D C B A 11
- 12. Prove the following expression by using truth table Y=XYZ+XY+XYZ 12
- 13. 13 X Y Z 𝑿 𝒁 XYZ 𝑿Y XY𝒁 XYZ+𝑿Y XYZ+𝑿Y+XY𝒁 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1
- 14. Prove the following expression by using truth table (X+Y)(X+Y)=X 14 X Y 𝒀 (X+Y) (X+𝒀) (X+Y)(X+𝒀) 0 0 1 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1
- 15. Simplify Simplify logical expression to its simplest form. ◦ F=𝑿YZ+𝑿Y𝒁+XZ 15
- 16. Simplify Draw the logic diagram for the following expression : 𝐹 = 𝐴 𝐵 + 𝐴𝐵 16
- 17. Simplify the following expression and draw a logic diagram for the simplified expression 𝑭 = 𝑨𝑩𝑪 + 𝑨𝑩𝑪 + 𝑨𝑩𝑪 + 𝑩𝑪 17 SOLUTION 𝑭 = 𝑨𝑩𝑪 + 𝑨𝑩𝑪 + 𝑨𝑩𝑪 + 𝑨𝑩𝑪 + 𝑩𝑪 theorem (5) 𝑭 = 𝑨 + 𝑨 𝑩𝑪 + 𝑨 𝑩𝑪 + BC + 𝑩𝑪 = 𝟏 𝑩𝑪 + 𝑨 𝑩 + B C + 𝑩𝑪 =𝑩𝑪 + 𝑨C + 𝑩𝑪 = 𝑩(𝑪 + 𝑪 ) +𝑨C F = 𝑩 + 𝑨𝑪
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- 19. Simplify the following expression and draw a logic diagram for the simplified expression 𝐹 = 𝑌 𝑊 𝑍 + 𝑌𝑊𝑍 + 𝑋𝑌 19 SOLUTION 𝐹 = 𝑌 𝑊 𝑍 + 𝑌𝑊𝑍 + 𝑋𝑌 𝐹 = 𝑌 𝑊 𝑍 + 𝑍 + 𝑋𝑌 𝐹 = 𝑌 𝑊 1 + 𝑋𝑌 𝐹 = 𝑌 𝑊 + 𝑋
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- 21. Simplify the following expression and draw a logic diagram for the simplified expression 𝐹 = 𝐴 𝐵𝐶 + 𝐴𝐵𝐶 + 𝐴𝐵𝐶 + 𝐴𝐵𝐶 21 SOLUTION 𝐹 = (𝐴 𝐵(𝐶 + 𝐶) + 𝐴𝐵𝐶) + 𝐴𝐵𝐶 = (𝐴 𝐵 + 𝐴𝐵𝐶) + 𝐴𝐵𝐶 = 𝐴 𝐵 + (𝐴𝐵𝐶 + 𝐴𝐵𝐶) = 𝐴 𝐵 + (𝐴𝐵 + 𝐴𝐵)C = 𝐴 𝐵 + (𝐴⨁𝐵)𝐶
- 22. Simplify the following expression and draw a logic diagram for the simplified expression 𝐹 = 𝐴 𝐵𝐶 + 𝐴𝐵𝐶 + 𝐴𝐵𝐶 + 𝐴𝐵𝐶 22 SOLUTION 𝐹 = 𝐴 𝐵 + (𝐴⨁𝐵)𝐶
- 23. Simplify the following expression and draw a logic diagram for the simplified expression 𝐹 = 𝐴 𝐵(𝐷𝐴 + 𝐶𝐷) + 𝐴𝐵(𝐴 + 𝐶) 23 SOLUTION 𝐹=𝐴 𝐵𝐷𝐴 + 𝐴𝐵𝐶𝐷 + 𝐴𝐵𝐴 + 𝐴𝐵𝐶 = 𝐴 𝐵𝐷𝐴 + 𝐴𝐵𝐶𝐷 + 𝐴𝐵𝐴 + 𝐴𝐵𝐶 =𝐴 𝐵 𝐷 + 𝐶𝐷 + 0 + 𝐴𝐵𝐶 identity 16 = 𝐴 𝐵 𝐷 + 𝐶 + 𝐴𝐵𝐶 =𝐴 𝐵𝐷 + 𝐴𝐵𝐶 + 𝐴𝐵𝐶 =(𝐴 𝐵 + 𝐴𝐵 )𝐶+ABD 𝐹 = (𝐴⨀𝐵)𝐶+ABD
- 24. Simplify the following expression and draw a logic diagram for the simplified expression 𝐹 = 𝐴 𝐵(𝐷𝐴 + 𝐶𝐷) + 𝐴𝐵(𝐴 + 𝐶) 24