1. Boolean Algebra Module M4.1 Section 5.1

2. Boolean Algebra and Logic Equations Switching Algebra Theorems Venn Diagrams

3. One-variable Theorems OR Version AND Version X # 0 = X X # 1 = 1 X & 1 = X X & 0 = 0 Note:Principle of Duality You can change # to & and 0 to 1 and vice versa

4. One-variable Theorems OR Version AND Version X # !X = 1 X # X = X X & !X = 0 X & X = X Note:Principle of Duality You can change # to & and 0 to 1 and vice versa

5. Two-variable Theorems Commutative Laws Unity Absorption-1 Absorption-2

6. Commutative Laws X # Y = Y # X X & Y = Y & X

7. Venn Diagrams X !X

8. Venn Diagrams XY X & Y

9. Venn Diagrams X # Y XY

10. Venn Diagrams !X & Y X Y

11. Unity !X & Y X Y X & Y (X & Y) # (!X & Y) = Y Dual: (X # Y) & (!X # Y) = Y

12. Absorption-1 X Y X & Y Y # (X & Y) = Y Dual: Y & (X # Y) = Y

13. Absorption-2 !X & Y X Y X # (!X & Y) = X # Y Dual: X & (!X # Y) = X & Y

14. Three-variable Theorems Associative Laws Distributive Laws

15. Associative Laws X # (Y # Z) = (X # Y) # Z Dual: X & (Y & Z) = (X & Y) & Z

16. Associative Law 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 X Y Z Y # Z X # (Y # Z) X # Y (X # Y) # Z X # (Y # Z) = (X # Y) # Z

17. Distributive Laws X & (Y # Z) = (X & Y) # (X & Z) Dual: X # (Y & Z) = (X # Y) & (X # Z)

18. Distributive Law - a

19. Distributive Law - b X & (Y # Z) = (X & Y) # (X & Z)

20. Generalized De Morgan’s Theorem NOT all variables Change & to # and # to & NOT the result -------------------------------------------- F = X & Y # X & Z # Y & Z F = !((!X # !Y) & (!X # !Z) & (!Y # !Z)) F = !(!(X & Y) & !(X & Z) & !(Y & Z))

22. NAND Gate

23. X Y X Z Y Z F F = X & Y # X & Z # Y & Z

24. Question The following is a Boolean identity: (true or false) Y # (X & !Y) = X # Y