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Application: Digital Logic Circuits

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.

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Application: Digital Logic Circuits

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  1. Application: Digital Logic Circuits Lecture 4 Section 1.4 Wed, Jan 25, 2006

  2. Logic Gates • Three basic logic 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.

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

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

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

  7. NOR-Gate • Output is 1 if both inputs are 0. • Output is 0 if either input is 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. Output Tables • An output table shows the output of the circuit for every possible combination of inputs.

  11. 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.

  12. Example: Designing a Circuit • Design a circuit for (pq).

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

  14. Example: Designing a Circuit • Design a circuit for (pq)  (qr).

  15. 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.

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

  17. 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.

  18. 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).

  19. 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 .

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

  21. 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.”

  22. 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?

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