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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

P

closed closed open open

Q

closed open closed open

State

on off off off – closed/on => T – open/off => F

P

T T F F F T

Q

T F F F

State

T F

Switching Table

• Switches in parallel

P

closed closed open open

Q

closed open closed open

State

on on on off – closed/on => T – open/off => F

P

T T F F F T

Q

T F T T

State

T F

Basic Digital Logic Gates

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).

Example

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.

Input

P 0 0 1 1 Q 0 1 0 1

Output

R X X X X

Example

P v Q ~(P ^ Q) P ^ Q 1 1 P 0 0

Input

Q 0 1 0 1 (P v Q) ^ ~(P ^ Q)

Output

R 0 1 1 0

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, ^ )

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

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

Example

1^1^1 v 1^0^1 v 1^0^0 P^Q^R v P^~Q^R v P^~Q^~R

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.

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.

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.)

NAND and NOR

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

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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