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The Logic of Compound Statements

Digital Logic Circuits

• • • Electrical circuits can be fashioned to mimic logic tables.

Types of switches: – open – closed Types of circuits: – series – parallel

• 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

• 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

• • • 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).

• 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

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

• • • • 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, ^ )

• • 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

• • 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

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

• • 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.

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

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