1. Boolean Algebra Discussion D2.2

2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams

3. George Boole English logician and mathematician Publishes Investigation of the Laws of Thought in 1854

4. 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

5. 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

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

7. Commutative Laws X + Y = Y + X X*Y = Y*X

8. Venn Diagrams X !X

9. Venn Diagrams XY X*Y

10. Venn Diagrams X + Y XY

11. Venn Diagrams X' * Y X Y

12. Unity X' * Y X Y X * Y (X * Y) + (X' * Y) = Y Dual: (X + Y)*(X' + Y) = Y

13. Absorption-1 X Y X & Y Y + (X * Y) = Y Dual: Y * (X + Y) = Y

14. Absorption-2 X' * Y X Y X + (X' * Y) = X + Y Dual: X * (X' + Y) = X * Y

15. Three-variable Theorems Associative Laws Distributive Laws

16. Associative Laws X + (Y + Z) = (X + Y) + Z Dual: X * (Y * Z) = (X * Y) * Z

17. 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

18. Distributive Laws X * (Y + Z) = (X * Y) + (X * Z) Dual: X + (Y * Z) = (X + Y) * (X + Z)

19. Distributive Law - a

20. Distributive Law - b X * (Y + Z) = (X * Y) + (X * Z)

21. Question The following is a Boolean identity: (true or false) Y + (X * Y') = X + Y

22. Absorption-2 X * Y' Y X Y + (X * Y') = X + Y