1. + CS 325: CS Hardware and Software Organization and Architecture Gates and Boolean Algebra Part 3

2. + Basic Laws of Boolean Algebra Boolean Algebra follows many algebra rules which can be used to make simpler circuits. Example: AB + ACThree gates = A(B + C), Distributive LawTwo gates NameAND FormOR Form Identity Law Null Law Idempotent Law Commutative Law Associative Law Distributive Law Absorption Law De Morgan’s Law

3. + Reduction using Boolean Laws

4. + Checking Reduction for Correctness 00000000 00100000 01000111 01100111 10000000 10101011 11011111 11111111

5. + 00000000 00100000 01000111 01100111 10000000 10101011 11011111 11111111

6. + Converting Boolean Functions to Standard SOP

7. +

8. + Sum-of-Products

9. + ABC 0000 0010 0101 0111 1000 1010 1101 1110

10. + Sum-of-Products – Another Example ABC 0000 0011 0101 0111 1000 1010 1100 1110

11. + Covered So Far: Logic Gate Diagrams Truth Tables Standard Sum-of-Products Sum of midterms Boolean functions And reduction using Boolean laws You should be able to derive one form using any other form! Truth Table  sum of midterms Reduced Boolean function  standard sum-of-products Logic Gate Diagram  Boolean function  reduced Boolean function  Logic Gate Diagram

12. + Product-of-Sums

13. + ABC 0000 0011 0101 0111 1001 1011 1101 1110

14. + Product-of-Sums - Another Example ABC 0001 0010 0101 0110 1000 1011 1101 1111

15. + Covered So Far: Logic Gate Diagrams Truth Tables Standard Sum-of-Products Standard Product-of-Sums Sum of midterms Boolean functions And reduction using Boolean laws

16. + Karnaugh Maps Special form of a given truth table. Useful for reducing logic functions into minimal Boolean expressions. B  A ABX 000 011 101 110 01 001 110

17. + Karnaugh Maps 2-Variables Convert the following 2-variable truth table to its Karnaugh map equivalent: B  A ABX 001 010 101 110 01 010 110

18. + Karnaugh Maps 3-Variables The following is an example of a 3-variable truth table converted to its Karnaugh map equivalent: C  AB ABCX 0000 0010 0101 0110 1001 1011 1100 1111 01 0000 0110 1011 1101

19. + Karnaugh Maps 3-Variables Convert the following 3-variable truth table to its Karnaugh map equivalent: C  AB ABCX 0001 0010 0100 0111 1001 1010 1100 1111 01 0010 0101 1010 1101

20. + Karnaugh Maps 4-Variables The following is an example of a 4-variable truth table converted to its Karnaugh map equivalent: CD  AB ABCDX 00000 00011 00101 00110 01000 01010 01100 01111 10000 10010 10101 10111 11000 11010 11100 11111 00011011 000110 010001 100011 110001

21. + Karnaugh Maps 4-Variables Convert the following 4-variable truth table to its Karnaugh map equivalent: CD  AB ABCDX 00000 00010 00100 00111 01001 01011 01100 01110 10001 10010 10100 10111 11001 11010 11100 11111 00011011 000001 011100 101001 111001