1. September 7 Notes
Boolean Algebra

2. Boolean Algebra
O and 1 are the only possible values for Boolean variables
The product of two Boolean variables x and y is written xy
The sum of two Boolean variables x and y is written x+y

3. Evaluating a Boolean expression
To determine the value of a Boolean expression we evaluate it using the following precedence rules:
What is in the parentheses first
Complement is a unary operator and takes precedence over the binary operators
And (product) takes precedence over Or (plus)

4. Page 702
Let B = {0,1}. Then Bn = {(x1, x2,…, xn) | xi  B for 1 ≤ I ≤ n} is the set of all possible n-tuples of 0s and 1s.
The variable x is called a Boolean variable if it assumes values only from B.
A function from Bn to B is called a Boolean function of degree n.
There are 2 to the 2n different functions of degree n.

5. (x1, x2,…, xn) is a n-tuple
Bn = {(x1, x2,…, xn) | xi  B for 1 ≤ I ≤ n}
B to the degree n is the set of tuples from x sub1 to x sub n such that each of the x sub I is an element of B for all of the Is between 1 and n

6. Any Boolean function can be represented by a Boolean sum of Boolean products of the variables and their complements
Every Boolean function can be represented using the three Boolean operators .,+,and –
A literal is a Boolean variable or its complement.
A minterm is a product of n variables

7. Two different Boolean expressions that represent the same function are called equivalent.
We can determine if two Boolean expressions are equivalent by evaluating them using truth tables.
Complement
Boolean sum
Boolean product

8. Boolean functions
A function of 2 Boolean variables will have 4 rows in its truth table
A function of 3 Boolean variables will have 8 rows in its truth table
A function of n Boolean variables will have 2n rows in its truth table.

9. Example 3 – p. 703
Example 3 on page 703 x y x'
x'y
f(x,y,z) = x'y 1

10. Example 1 - p. 709 Example 1 p 709 x y z F G 1 F(x,y,z) = xy'z
F(x,y,z) = xy'z
G(x,y,z) = xyz' + x'yz'
F has 1 minterm
G has 2 minterms

11. Example 3 - p. 710 Example 3 p 710 x y z x+y z' (x+y)z' 1
F(x,y,z) = (x+y)z'