1. Boolean Algebra Discussion D6.1 Appendix G

2. Boolean Algebra and Logic Equations George Boole - 1854 Boolean 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

23. Venn Diagrams and Minterms

24. xyz + xyz + xyz = xz + xy