1. Computer Science 210 Computer Organization
Introduction to
Boolean Algebra

2. George Boole English mathematician (1815-1864) Boolean algebra Logic
Set Theory
Digital circuits
Programming: Conditions in while and if statements

3. Boolean Constants
In Boolean algebra, there are only two constants, true and false
Boolean constant
Binary digit
State of a switch
Voltage level
true 1 On
+5V
false
Off 0V

4. Boolean Variables
Boolean variables are variables that store values that are Boolean constants.
Let A be true
Let B be false
Etc.

5. Boolean Operator AND
If A and B are Boolean variables (or expressions) then
A AND B
is true if and only if both A and B are true.

6. Boolean Operator AND
If A and B are Boolean variables (or expressions) then
A AND B
is false if and only if either A or B are false or they’re both false.

7. Boolean Operator AND
We denote the AND operation like multiplication in ordinary algebra:
AB or A.B

8. Boolean Operator OR
If A and B are Boolean variables (or expressions) then
A OR B
is true if and only if at least one of A and B is true.

9. Boolean Operator OR
If A and B are Boolean variables (or expressions) then
A OR B
is false if and only if both A and B are false.

10. Boolean Operator OR
We denote the OR operation like addition in ordinary algebra:
A+B

11. Boolean Operator NOT
If A is a Boolean variable (or expression) then NOT A
has the opposite value from A.

12. Boolean Operator NOT
We denote the NOT operation by putting a bar over the variable (or expression)
_ A
Or use tilde (~) when bar is not available: ~A

13. Boolean Expressions
As with ordinary algebra, a Boolean expression is a well-formed expression made from
Boolean constants
Boolean variables
Operators AND, OR and NOT
Parentheses
Example: __ ____ AB + (A+C)B

14. Evaluating a Boolean expression
At any time, the value of a BE can be computed using the current values of the variables. __
AB + (CD)
Let A = true
Let B = false
Let C = true
Let D = false
Then the resulting value is true

15. Operator precedence NOT comes first, then AND, and finally OR
(Like arithmetic negation, product, and addition)
A + BC is not the same as
(A + B)C

16. Evaluating a Boolean expression
Unlike ordinary algebra, for a BE, there are only finitely many possible assignments of values to the variables; so, theoretically, we can make a table, called a truth table, that shows the value of the BE for every possible set of values of the variables.
For convenience, use
0 = false
1 = true

17. Truth Table for AND A B AB 1

18. Truth Table for OR A B
A+B 1

19. Truth Table for NOT A _ 1

20. Filling in a Truth Table
If there are N variables, there are 2N possible combinations of values
Thus, there are 2N rows in the truth table
Fill in the values by counting up from 0 in binary

21. Example
Construct a truth table for
_ ___ E = AB + (A+C)B

22. _ ___ E = AB + (A+C)B A B C 1
Assign the values of the variables first

23. _ ___ E = AB + (A+C)B Then add columns for each operation A B C 1 _ B1 _ B 1
Then add columns for each operation

24. _ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1

25. _ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1
A+C 1

26. _ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1
A+C 1
___
(A+C) 1

27. _ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1 ___1 _ B 1 _ AB 1
A+C 1
___
(A+C) 1
___
(A+C)B 1

28. _ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1 ___1 _ B 1 _ AB 1
A+C 1
___
(A+C) 1
___
(A+C)B 1 E 1