1. Application: Digital Logic Circuits
Lecture 4
Section 1.4
Mon, Jan 17, 2005

2. Logic Gates Three basic logic gates Two other gates AND-gate OR-gate
NOT-gate
Two other gates
NAND-gate (NOT-AND)
NOR-gate (NOT-OR)

3. AND-Gate Output is 1 if both inputs are 1.
Output is 0 if either input is 0. p q
Output 1

4. OR-Gate Output is 1 if either input is 1.
Output is 0 if both inputs are 0. p q
Output 1

5. NOT-Gate Output is 1 if input is 0. Output is 0 if input is 1. p

6. NAND-Gate Output is 1 if either input is 0.
Output is 0 if both inputs are 1. p q
Output 1

7. NOR-Gate Output is 1 if both inputs are 0.
Output is 0 if either input is 1. p q
Output 1

8. Disjunctive Normal Form
A logical expression is in disjunctive normal form if
It is a disjunction of clauses.
Each clause is a conjunction of variables and their negations.
Each variable or its negation appears in each clause exactly once.

9. Examples: Disjunctive Normal Form
p  q  (p  q)  (p  q)  (p  q).
p  q  (p  q)  (p  q).
p | q  (p  q)  (p  q)  (p  q).
p  q  p  q.

10. Conjunctive Normal Form
A logical expression is in conjunctive normal form if
It is a conjunction of clauses.
Each clause is a disjunction of variables and their negations.
Each variable or its negation appears in each clause exactly once.

11. Examples: Conjunctive Normal Form
p  q  p  q.
p  q  (p  q)  (p  q).
p | q  p  q.
p  q  (p  q)  (p  q)  (p  q).

12. Output Tables
An output table shows the output of the circuit for every possible combination of inputs.
Inputs
Output 1

13. Designing a Circuit Write an output table for the circuit.
Write the expression in disjunctive normal form.
Simplify the expression as much as possible.
Write the circuit using AND-, OR-, and NOT-gates.

14. Example: Designing a Circuit
Design a circuit for (p  q).
Inputs
Output p q 1

15. Example: Designing a Circuit
(p  q) is equivalent to p  q.
Draw the circuit using an AND-gate and a NOT-gate.

16. Example: Designing a Circuit
Design a circuit for (p  q)  (q  r).
Inputs
Output p q r 1

17. Example: Designing a Circuit
(p  q)  (q  r) is equivalent to
(p  q  r)  (p  q  r)  (p  q  r).
Does this simplify?
In any case, we can draw a circuit, although it may not be optimal.

18. Puzzle Three men apply for a job.
They are equally well qualified, so the employer needs a way to choose one.
He tells them
“On the forehead of each of you I will put either a red dot or a blue dot.”
“At least one of you will have a red dot.”
“The first one who can tell me the color of the dot on his forehead gets the job.”

19. Puzzle The employer proceeds to put a red dot on each man’s forehead.
After a few moments, one of them says, “I have a red dot.”
How did he know?