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

29. Designing a Circuit from a Truth Table
The problem reduces to this:
Given a truth table with all values for inputs.
And given a column of values for the output.
Find a Boolean expression that gives the column.
If we can do this, we can get the circuit from the Boolean expression.
The Sum of Products algorithm finds the expression.

30. Sum of Products Algorithm
Identify each row of the output that has a 1.
For each such row
Make a product of all the input variables.
Put bar over each variable with a 0 in this row.
Make a sum of all of these product terms.
Then, simplify the expression if possible.