Application: Digital Logic Circuits Lecture 5 Section 1.4 Wed, Jan 24, 2007

Logic Gates Three basic logic 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. pqOutput

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

NOT-Gate Output is 1 if input is 0. Output is 0 if input is 1. pOutput 10 01

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

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

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.

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

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 pq

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 pqr

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.

Example: Designing a Circuit Design a logic circuit for (p  q)  (  q   r)  r.

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).

Conjunctive Normal Form To write an expression in CNF, Write the output table (truth table). Follow the procedure for writing the expression in DNF, except Reverse the rolls of 0 and 1 and  and .

Example: Using CNF Re-do the previous example (p  q)  (  q   r)  r. using the conjunctive normal form.

The Red Dot-Blue Dot 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.”

The Red Dot-Blue Dot 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?