1. Boolean Algebra
How gates get picked

2. Boolean Arithmetic Boolean ≠ Binary Boolean addition:
Boolean – 1/0 only, no places
Boolean addition:

3. Boolean Arithmetic
Boolean addition:
+ means OR

4. Boolean Multiplication

5. Boolean Multiplication
Multiplication is AND

6. Boolean Variables Capital letter used for variables Inverted A, B, C…
A or A'

7. Identities
Additive:

8. Identities
Multiplicative:

9. Double Complement
Two negations cancel:
A = A

10. Break …

11. Commutative Property
Order of variables not important:

12. Associative Property
+ and · associative
Just like "normal" algebra

13. Distributive
Can distribute/factor
Just like "normal" algebra

14. Theorem 1
A + AB = A
I have A Or I have A and B
I really just need A

15. Theorem 2 A + A B = A + B I have A Or I have B but not A
I need A or B (A satisfies first term - only care about second if first is not satisfied)

16. Theorem 3 (A + B)(A + C) = A + BC I have A or B AND I have A or C
I need A or both B and C

17. Example:
A + AC + BC A(1 + C) + BC A(1) + BC A + BC

18. Example:
AB + AB' A(B + B') A(1) A
Note : A' = A

19. Example:
A + A'B' + B' A + B'(A' + 1) A + B'(1) A + B'

20. Example:
A + B + A'
(A + A') + B
+ B 1

21. Example:
A(B +AB) + AC AB + AAB + AC AB + AB + AC AB + AC A(B + C)

22. Truth Table Truth table defines Boolean function
When in doubt, check the truth table…

23. Proof By Exhaustion
Proof by exhaustion : prove equivalence by comparing truth tables
Ex: AB = A + B A B AB 1 A B
A + B 1

24. Not The Same AB != A · B A + B != A + B A B AB 1 A B A · B 1 A B A + B1 A B
A · B 1 A B
A + B 1 A B
A+B 1

25. DeMorgan's Theorems A+B = A ∙ B AB = A + B
If you don't have both of A and B …you do not have A or do not have B
A+B = A ∙ B
If you don't have either of A or B …you do not have A and you do not have B

26. DeMorgan's
Break up solid bar by switching operation:

27. Samples
AB + A B = A + B + A B DeMorgan's = A + A B + B Commutative = A (1 + B) + B Distributive = A (1) + B 1+ anything = 1 = A + B 1 · anything = self

28. Samples
( A +B) ·A = ( A · B )·A DeMorgan's = (A· B )·A Double negative cancel = A· B ·A Associative = A·A· B Commutative = A· B Anything · self = self