Digital Logic Circuits - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

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Chapter 1

The Logic of Compound Statements

Section 1.

Digital Logic Circuits

Switching Table

• Switches in series

– closed/on => T

– open/off => F

P Q State closed closed on closed open off open closed off open open off P Q State T T T T F F F T F F F F

Switching Table

• Switches in parallel

– closed/on => T

– open/off => F

P Q State T T T T F T F T T F F F P Q State closed closed on closed open on open closed on open open off

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

Example

Input Output P Q R 0 0 0 0 1 1 1 0 1 1 1 0 P v Q P ^ Q

~(P ^ Q)

(P v Q) ^ ~(P ^ Q)

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, ^ )

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^ P^Q^R v P^~Q^R v P^~Q^~R

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 (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 |)