1. Microprocessors vs. Custom Digital Circuits
Designers that work with digital phenomena often buy an off-the-shelf microprocessor and program it.
Microprocessors are readily available, inexpensive, easy to program, and easy to reprogram
Why would anyone ever need to design new digital circuits?

2. Microprocessors vs. Custom Digital Circuits
Designers that work with digital phenomena often buy an off-the-shelf microprocessor and program it.
Microprocessors are readily available, inexpensive, easy to program, and easy to reprogram
Why would anyone ever need to design new digital circuits?
Microprocessors are sometimes:
Too slow;
Too big;
Consume too much power;
Too costly

3. Combinatorial Logic Circuits
A digital circuit whose output depends solely on the present combination of input values is called a combinatorial circuit
Logic gates – building blocks of logic circuits
AND OR NOT
Boolean Algebra
Boolean algebra is a branch of mathematics that uses variables whose values can only be 1 or 0 (“true” or “false”, respectively) and whose operators, like AND, OR, NOT, operate on such variables and return 1 or 0.
We can build circuits by doing math

4. Example: Seatbelt warning light
Logic Gates
Truth Tables
Example: Seatbelt warning light
Design a system for an automobile that illuminates a warning light whenever the driver’s seatbelt is not fastened, and the key is in the ignition
Boolean equation:
w = NOT(s) AND k

5. Notation and Terminology
Operators:
NOT(a) is typically written as a’
a OR b is typically written as a + b
a AND b is typically written as a * b (or a b)
w = NOT(s) AND k = s’k
Precedence rule:
Expression in parentheses
AND, NOT OR
w = (a + b) * (c’) + d

6. Properties of Boolean Algebra
Commutative
a + b = b + a
a * b = b * a
Distributive
a * (b + c) = a * b + a * c
a + (b * c) = (a + b) * (a + c)
Associative
(a + b) + c = a + (b + c)
(a * b) * c = a * (b * c)
Identity
0 + a = a + 0 = a
1 * a = a * 1 = a
Complement
a + a’ = 1
a * a’ = 0

7. Additional Properties
Null elements
a + 1 = 1
a * 0 = 0
Idempotent Law
a + a = a
a * a = a
Involution Law
(a’)’ = a
De Morgan’s Law
(a + b)’ = a’ b’
(a b)’ = a’ + b’
Example: Simplification of an automatic sliding door system
f = h c’ + h’ p c’

8. Additional Properties
Null elements
a + 1 = 1
a * 0 = 0
Idempotent Law
a + a = a
a * a = a
Involution Law
(a’)’ = a
De Morgan’s Law
(a + b)’ = a’ b’
(a b)’ = a’ + b’
Example: Simplification of an automatic sliding door system
f = h c’ + h’ p c’
f = c’ (h + p)

9. Boolean Functions Converting a truth table to an equation
Boolean function is a mapping of each possible combination of input values to either 0 or 1.
Boolean function can be represented as an equation, a circuit, and as a truth table.
Converting a truth table to an equation
F = a b + a’
F = a’ b’ + a’ b + a b
For any function, there may be many equivalent equations, and many equivalent circuits, but there is only one truth table!