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Explore the fundamentals of logic gates including AND, OR, NOT, NAND, and NOR gates. Learn about Disjunctive Normal Form, Output Tables, and Designing Circuits using different gate combinations. Delve into Conjunctive Normal Form and solve logic puzzles.
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Application: Digital Logic Circuits Lecture 4 Section 1.4 Wed, Jan 25, 2006
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.
OR-Gate • Output is 1 if either input is 1. • Output is 0 if both inputs are 0.
NOT-Gate • Output is 1 if input is 0. • Output is 0 if input is 1.
NAND-Gate • Output is 1 if either input is 0. • Output is 0 if both inputs are 1.
NOR-Gate • Output is 1 if both inputs are 0. • Output is 0 if either input is 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 • pq (pq) (pq) (pq). • p q (pq) (pq). • p | q (pq) (p q) (pq). • pqpq.
Output Tables • An output table shows the output of the circuit for every possible combination of inputs.
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 (pq).
Example: Designing a Circuit • (pq) is equivalent to pq. • Draw the circuit using an AND-gate and a NOT-gate.
Example: Designing a Circuit • Design a circuit for (pq) (qr).
Example: Designing a Circuit • (pq) (qr) is equivalent to (pqr) (pqr) (pqr). • 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 (pq) (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 • pqpq. • p q (pq) (pq). • p | qpq. • pq (pq) (pq) (pq).
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 (pq) (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?