September 7 Notes Boolean Algebra Boolean Algebra O

September 7 Notes Boolean Algebra

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

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)

Page 702 • Let B = {0, 1}. Then Bn = {(x 1, x 2, …, xn) | xi B for 1 ≤ I ≤ n} is the set of all possible n-tuples of 0 s and 1 s. • 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 2 n different functions of degree n.

• (x 1, x 2, …, xn) is a n-tuple • Bn = {(x 1, x 2, …, xn) | xi B for 1 ≤ I ≤ n} • B to the degree n is the set of tuples from x sub 1 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

• 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

• 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

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 2 n rows in its truth table.

Example 3 – p. 703 Example 3 on page 703 f(x, y, z) = x'y x y x' x'y 0 0 1 1 1 1 0 0 0 1 1 0 0

Example 1 - p. 709 Example 1 p 709 x y z F G 1 1 1 0 0 1 F(x, y, z) = xy'z 1 0 1 1 0 G(x, y, z) = xyz' + x'yz' 1 0 0 0 1 1 0 0 F has 1 minterm 0 1 0 0 1 G has 2 minterms 0 0 1 0 0 0 0

Example 3 - p. 710 Example 3 p 710 F(x, y, z) = (x+y)z' x y z x+y z' (x+y)z' 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 0