1. Boolean Logic

2. Boolean Operators (T/F)x y
x AND y F T x
NOT x F T x y
x OR y F T x y
x XOR y F T

3. Boolean Operators (1/0) x y x AND y 1 x NOT x 1 x y x OR y 1 x y1 x
NOT x 1 x y
x OR y 1 x y
x XOR y 1

4. Boolean Operators Symbols
NOT
ā (overbar), a’, ~a
AND
· (mult. dot) OR +
XOR
 (plus sign with circle around it)

5. Boolean Expressions Follows a logical order of operations Example:
NOT operators
Parentheses
AND OR
Example:
x + y·z

6. Truth Tables
Write out table of all possible combinations of truth values
Evaluate the boolean expression for all combinations
Example
x + y·z x y z
x + y·z F T

7. Example
What is the truth table for: ~x + y? x y
~x + y F T

8. Another Example
What is the truth table for: x·(~y)? x y
x·(~y) F T

9. Your Turn
What is the truth table for the boolean expression: x + ~y + z?

10. Simplifying Boolean Expressions
Commutative laws
A + B = B + A
A · B = B · A
Associative laws
A + (B + C) = (A + B) + C
A · (B · C) = (A · B) · C
Distributive laws
A · (B + C) = A · B + A · C
A + (B · C) = (A + B) · (A + C)

11. Simplifying Boolean Expressions
Tautology laws
A · A = A
A + A = A
A + ~A = 1
A · ~A = 0
Absorption Law
A + (A · B) = A
A · (A + B) = A

12. Simplifying Boolean Expressions
Identities
0 · A = 0
0 + A = A
A + 1 = 1
1 · A = A
A = A
Complement
A + ~A · B = A + B

13. Examples
A + A + A + A = A
Using the Tautology law

14. A Bigger Example Simplify ~A · B + A · ~B + ~A · ~B
~A · B + (A · ~B + ~A · ~B)  Associative
~A · B + (~B · (A + ~A))  Distributive
~A · B + ~B & Tautology
~A + ~B  Complement
Verify with a truth table!

15. Practice
Show that A + B · C = (A + B) · (A + C) is true using a truth table.

16. Practice
Show that A + ~A · B = A + B

17. Practice Simplification
Simplify A + AB + ~B and verify with a truth table

18. De Morgan’s Laws ~(A · B) = ~A + ~B ~A · ~B = ~(A+B) Take a term
NOT the individual members of the term
A · B
Change the operator i.e. · to +, or + to ·
A + B
NOT the entire term
~(A+B)

19. De Morgan’s Law Example
f = ~A · ~B + (~A + ~B)
= ~~( ~A · ~B + (~A + ~B) )  NOT NOT
= ~( (A + B) · ~(~A + ~B) )  De Morgan’s
= ~( (A + B) · (A·B) )  De Morgan’s
= ~( A·(A·B) + B·(A·B) )  Distributive
= ~( A·B + A·B )  Tautology
= ~(A·B)  Tautology