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Chapter 1

Chapter 1. The Logic of Compound Statements. Section 1.4. Digital Logic Circuits. Digital Circuits. Electrical circuits can be fashioned to mimic logic tables. Types of switches: open closed Types of circuits: series parallel . Switching Table. Switches in series closed/on => T

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Chapter 1

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  1. Chapter 1 The Logic of Compound Statements

  2. Section 1.4 Digital Logic Circuits

  3. Digital Circuits • Electrical circuits can be fashioned to mimic logic tables. • Types of switches: • open • closed • Types of circuits: • series • parallel

  4. Switching Table • Switches in series • closed/on => T • open/off => F

  5. Switching Table • Switches in parallel • closed/on => T • open/off => F

  6. Basic Digital Logic Gates

  7. Combinational Circuits • Combinational circuits are composed of one or more basic gates where the output of the circuit is based on the input at that instant in time. • Rules of Combinational Circuits • Never combine two input wires. • A single input wire can be split and used as input for two separate gates. • An output wire can be used as input. • No output of a gate can feedback into that gate. • Sequential circuits are circuits that include feedback. Their output depends on previous input. These circuits are used to build circuits that can remember (memory circuits).

  8. Example

  9. Input-Output Table • Input-output table is a truth table for a combinational circuit. It shows the output of the circuit given a set of inputs.

  10. Example P v Q (P v Q) ^ ~(P ^ Q) ~(P ^ Q) P ^ Q

  11. Boolean • A combinational circuit can be expressed as a Boolean expression. • George Boolean was an English mathematician who founded symbolic logic. • Boolean variable is a variable that has only two possible values (T/F, on/off, 1/0). • Boolean expression is composed of Boolean variables and connectives (~, v, ^ )

  12. Boolean Expression Circuits • A Boolean expression can be converted to a combinational digital logic circuit by using the Boolean variables as inputs and matching the connectives (~, v, ^) with their gate equivalent (NOT, OR, AND). • Example • (~P ^ Q) v ~Q

  13. Circuit from I/O Table • A circuit can be constructed from any I/O table. • A circuit constructed in this form will be composed of a set of AND gates connected by OR gates. R^S v ~R^S v R^~S

  14. Example 1^1^1 v 1^0^1 v 1^0^0 P^Q^Rv P^~Q^Rv P^~Q^~R

  15. Equivalent Circuits • Two circuits are equivalent if there I/O tables are equivalent. • As with logic expressions, digital circuits may be simplified through logic theorem 1.1.1, aka Boolean Algebra.

  16. Example • ((P ^ ~Q) V (P ^ Q)) ^ Q • (P ^ (~Q V Q)) ^ Q (distributive) • (P ^ (Q v ~Q)) ^ Q (commutative) • (P ^ t) ^ Q (negation) • P ^ Q (identity) • Inspection of the I/O table reveals the simplified circuit.

  17. NAND and NOR Gates • NAND or NOR gates can be used to simplify a circuit as they are primitive gates, i.e. all gates can be built from them. (NOT, AND, OR, XOR, etc.)

  18. NAND and NOR • NAND • logic symbol is (Sheffer Stroke) | • P|Q  ~(P ^ Q) • NOR • logic symbol is (Peirce Arrow)  • PQ  ~(P v Q)

  19. NAND (Sheffer Stroke) Example • Show that the Sheffer Stroke (NAND) can be used to implement ~ (NOT) • ~P  P | P • ~P  ~(P ^ P) (idempotent) •  P | P (definition of |)

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