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

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)

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

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

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

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

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

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.

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.

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.

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).

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

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.

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

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

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

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.

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.”

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?