in this lecture: lecture 4: boolean...

E1.2 Digital Electronics 1 4.1 23 October 2008

Lecture 4: Boolean Algebra

Dr Pete Sedcole

Department of E&E Engineering

Imperial College London

http://cas.ee.ic.ac.uk/~nps/

(Floyd 4.2 – 4.5, 5.2 – 5.4)

(Tocci 3.10 – 3.14)


In this lecture:

• Theorems and rules in Boolean algebra

• DeMorgan’s Theorems

• Universality of NAND and NOR gates

• Active LOW & active HIGH

• Digital Integrated Circuits


Laws of Boolean Algebra

• Commutative Laws

• Associative Laws

• Distributive Law


Commutative Laws of Boolean Algebra

A + B = B + A

A • B = B • A


Associative Laws of Boolean Algebra

A + (B + C) = (A + B) + C

A • (B • C) = (A • B) • C


Distributive Laws of Boolean Algebra

A • (B + C) = A • B + A • C

A(B + C) = AB + AC


Rules of Boolean Algebra


Rules of Boolean Algebra 1–4

OR Truth Table

AND Truth Table

Rule 1

Rule 2

Rule 3

Rule 4



OR Truth Table

AND Truth Table

Rule 5

Rule 6

Rule 7

Rule 8E1.2 Digital Electronics 1 4.10 23 October 2008


AND Truth Table OR Truth Table

Rule 9

Rule 10: A + AB = A


Rules of Boolean Algebra 11–12 Rule 11: A + AB = A + B

Rule 12: (A + B)(A + C) = A + BC


Simplifying Boolean expressions: examples

• Simplify

• Simplify

• Simplify

BAy =

DBADBAy +=

( )( )BABAz ++=

BCDAACDx +=

Bz =

BCDACDx +=


Review questions

CAy =

CABCAy +=

DCBADCBAy +=

ABDDAy +=

DBAy =

BDDAy +=

• Simplify

• Simplify

• Simplify


DeMorgan’s Theorems

• Theorem 1:

• Theorem 2:

( ) yxyx ⋅=+

( ) yxyx +=⋅

Remember:

“break the bar, change the operator”

• DeMorgan’s Theorems are very useful in digital circuit design

• It allows AND gates to be exchanged with ORs by using inverters

• DeMorgan’s Theorems can be extended to any number of variables


Example of DeMorgan’s Theorem

QPYX

QPYXF

+++=

+= .. ← 2 NAND plus 1 OR

← 1 OR with some

input inverters


Example

• Simplify the expression

to one having only single variables inverted.

( ) ( )DBCAz +⋅+=

DBCAz +=


Using DeMorgan’s Theorems on logic gates


Using DeMorgan’s Theorems on logic gates


Example

• Determine the output expression for the circuit below

and simplify it using DeMorgan’s Theorem


Review questions

• Using DeMorgan’s Theorem, convert the expressions on

the left into ones that have only single-variable inversions:

( ) CBAz ⋅+=

QTSRy +=

DCBAy ++=

CBAz +=

QTSRy )( ++=

)( DCBAy +=


Universality of NAND gates

Any logic circuit can be implemented using only NAND gates


Universality of NOR gates

Any logic circuit can be implemented using only NOR gates


Basic Characteristics of Digital ICs

• Digital Integrated Circuits (ICs or chips): a collection of resistors, diodes and transistors fabricated on a single piece of semiconductor material

• The semiconductor is usually silicon (Si)

• Dual-in-line package(DIP) is a common type of packaging

7


Example DIP logic packages


The logic for x can be

constructed from 2 DIPs

It can also be built from

NAND gates only – using

only one DIP


Alternative logic-gate representations


Bubble notation

• A bubble (small circle) on an input or output of a gate is equivalent to an inverter

• To find the alternative logic symbol from the standard

symbol:

– Add bubbles to inputs and outputs that don’t have

them already

– Remove bubbles that are already there

– Swap AND ↔ OR


Things to notice:

• Alternative symbols can be extended to gates with any number of inputs

• The standard symbols never have bubbles on their inputs

• The alternative ones always have bubbles on their inputs

• The standard and alternative symbols represent the

same physical circuit


Active HIGH and active LOW

• An input or output that has a bubble on it is said to be

active-high

• An input or output that has no bubble on it is said to be

active-low


Interpretation of two NAND gate symbols


Interpretation of two NOR gate symbols


Equivalency of three circuits


Better diagram

Which circuit diagram should be used?

• Whenever possible, choose gate symbols so that nonbubble outputs are connected to nonbubble inputs, and bubble outputs are connected to bubble inputs

• This makes sure that the active-high / active-low interpretations are consistent

Output

active-low

Input

active-high


Active-high and active-low representations


More terminology

• When a logic signal is in its active state, it is said to be

asserted

• When a logic signal is not in its active state, it is said to be inactive or unasserted

• An overbar can be used to emphasise when a signal is active-low

1

01

A is asserted

Z is unasserted

Z


Summary

• How can we represent basic logic functions?

– Logical statements in our own language

– Truth tables

– Traditional graphic logic symbols

– Boolean algebra expressions

– Timing diagrams


Summary• Boolean algebra: a mathematical tool that is very useful for

analysing and designing digital circuits

– Using the rules and theorems of Boolean algebra can lead to a simpler way of implementing the circuit

• AND, OR, NOT: basic Boolean operations

– OR: HIGH when any input is HIGH

– AND: HIGH when all inputs are HIGH

– NOT: output is opposite of input

• NAND and NOR: can be used to implement any of the basic Boolean operations

– NOR: like OR with the output connected to an inverter

– NAND: like AND with the output connected to an inverter

in this lecture: lecture 4: boolean...

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