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Boolean Algebra

Summary


Digital Circuits (Chips/ICs/Transistors)


Basic Logic gates


Boolean Algebra, Basic Rules and Identities


Logic Simplification using Boolean Algebra


De Morgan’s Theorem and its Application in
Logic Simplification


Karnaugh Maps


Logic simplification using
K’Maps


Implicants

Chips/ICs


Our world is full of integrated circuits (ICs)


We can found ICs starting from Microprocessor
in our computer to almost every modern
electrical device such as Car, TV, CD Player,
Cell Phone, Electric oven, washing machine etc.


Made from different electrical components such
as transistors, resistors, capacitors and diodes

Gordon E. Moores’s Laws


Transistors per an IC:

Doubling of the number of
transistors on integrated circuits every two years (at
least for 1 more decade)


Cost per transistor:

As the size of transistors has
decreased, the cost per transistor has decreased as
well


Computing performance per unit cost:

As the size of
transistors shrinks, the speed at which they operate
increases



Transistor count

is the most common measure of chip complexity.

Processor

Transistor count

Date of introduction

Manufacturer

Intel 4004

2300

1971

Intel

Intel 8008

2500

1972

Intel

Intel 8080

4500

1974

Intel

Intel 8088

29

000

1979

Intel

Intel 80286

134

000

1982

Intel

Intel 80386

275

000

1985

Intel

Intel 80486

1

200

000

1989

Intel

Pentium

3

100

000

1993

Intel

AMD K5

4

300

000

1996

AMD

Pentium II

7

500

000

1997

Intel

AMD K6

8

800

000

1997

AMD

Pentium III

9

500

000

1999

Intel

AMD K6
-
III

21

300

000

1999

AMD

AMD K7

22

000

000

1999

AMD

Pentium 4

42

000

000

2000

Intel

Barton

54

300

000

2003

AMD

AMD K8

105

900

000

2003

AMD

Itanium 2

220

000

000

2003

Intel

Itanium 2

with 9MB cache

592

000

000

2004

Intel

Cell

241

000

000

2006

Sony
/
IBM
/
Toshiba

Core 2 Duo

291

000

000

2006

Intel

Core 2 Quad

582

000

000

2006

Intel

G80

681

000

000

2006

NVIDIA

POWER6

789

000

000

2007

IBM

Dual
-
Core
Itanium 2

1 700

000

000

2006

Intel

Quad
-
Core Itanium
Tukwila
[1]

2 000

000

000

2008

Intel

Boolean algebra



There are only two possible values for any quantity and
for any arithmetic operation 1 or 0


It does not matter how many or few terms we add
together, either.




Addition in Boolean Algebra


Is
not same

as real
-
number algebra

Multiplication in Boolean Algebra


Is
same

as in real
-
number algebra. Anything multiplied
by 0 is 0, and anything multiplied by 1 remains
unchanged



Logic Gates


A logic gate is an elementary building block of a digital
circuit.


There are AND, OR, NOT, NAND, NOR, EXOR and
EXNOR gates.


Most logic gates have two inputs and one output.


At any given moment, every terminal is in one of the two
binary conditions
low

(0) or
high

(1), represented by
different voltage levels.


The logic state of a terminal can, and generally does,
change often, as the circuit processes data.


In most logic gates, the low state is approximately zero
volts (0 V), while the high state is approximately five volts
positive (+5 V).


Boolean Addition corresponds to the logical
function of an "OR" gate



Boolean addition corresponds to the logical
function of an “AND" gate


Boolean compliment corresponds to the
logical function of a “NOT" gate


Boolean Identities


The sum of anything and zero is the same as the
original "anything."


This identity is no different from its real
-
number
algebraic equivalent


Boolean Identities


The sum of anything and one is one


Different from normal algebra


Boolean Identities


Adding A and A together


Is same as connecting both inputs of an OR gate to each other and
activating them with the same signal


Boolean Identities


The sum of a variable and its complement is 1



Boolean Identities


Just as there are four Boolean additive identities (A+0, A+1, A+A,
and A+A'), so there are also four multiplicative identities: Ax0, Ax1,
AxA, and AxA'. Of these, the first two are no different from their
equivalent expressions in regular algebra:


Boolean Identities


The third multiplicative identity: The product of a
Boolean quantity and itself is the original quantity,


since 0 x 0 = 0 and 1 x 1 = 1


Boolean Identities


The fourth multiplicative identity: The product of a
variable and its complement is 0



Boolean Identities (Summary)


Boolean Identities


Double complement
: a variable inverted twice.
Complementing a variable twice (or any even number of
times) results in the original Boolean value.


Laws of Boolean Algebra


The commutative law/property tells that, we can reverse the order
of variables that are either added together or multiplied together
without changing the truth of the expression



Laws of Boolean Algebra


Associative law tells that, we can associate groups of added or multiplied
variables together with parentheses without altering the truth of the
equations



Laws of Boolean Algebra


Distributive property: The Boolean expression formed by the product of a
sum, and in reverse shows how terms may be factored out of Boolean
sums
-
of
-
products


Basic Boolean Algebraic properties


Commutative, Associative, and Distributive


Boolean Rules


Boolean algebra finds its most practical use in the
simplification of logic circuits.


If we translate a logic circuit's function into symbolic
(Boolean) form, and apply certain algebraic rules to the
resulting equation to reduce the number of terms and/or
arithmetic operations, the simplified equation may be
translated back into circuit form for a logic circuit
performing the same function with fewer components.


If a equivalent function may be achieved with fewer
components, the result will be increased reliability and
decreased cost of manufacture.


Boolean Rules


This rule may be proven symbolically by factoring an "A" out of the
two terms, then applying the rules of A + 1 = 1 and 1A = A to
achieve the final result:

Boolean Rules


Boolean Rules


Proving using truth table

Boolean Rules


Simplification of a product
-
of
-
sums expression


Boolean Rules (Summary)


DeMorgan's Theorems



DeMorgan's Theorems


Reducing the expression (A + (BC)')' to A’BC using DeMorgan's Theorems


DeMorgan's Theorems


Reducing the expression (A + (BC)')' to A’BC using DeMorgan's Theorems


DeMorgan's Theorems


Maintaining the grouping implied by the complementation bars for the
expression is crucial to obtaining the correct answer


DeMorgan's Theorems


Applying the principles of DeMorgan's theorems to the simplification of a gate circuit


Label the outputs of the first NOR gate and the NAND gate


Finally, write an expression (or pair of expressions) for the last NOR gate


DeMorgan's Theorems


Reduce the expression using the identities, properties, rules, and theorems
(DeMorgan's) of Boolean algebra



DeMorgan's Theorems (Review)


DeMorgan's Theorems describe the equivalence between gates with
inverted inputs and gates with inverted outputs. Simply put, a NAND
gate is equivalent to a Negative
-
OR gate, and a NOR gate is
equivalent to a Negative
-
AND gate.


When "breaking" a complementation bar in a Boolean expression,
the operation directly underneath the break (addition or
multiplication) reverses, and the broken bar pieces remain over the
respective terms.


It is often easier to approach a problem by breaking the longest
(uppermost) bar before breaking any bars under it. You must
never

attempt to break two bars in one step!


Complementation bars function as grouping symbols. Therefore,
when a bar is broken, the terms underneath it must remain grouped.
Parentheses may be placed around these grouped terms as a help
to avoid changing precedence.

Karnaugh Maps


Applying Boolean algebra can be awkward in order to simplify
expressions


It is laborious and requires remembering all the laws


The Karnaugh map provides a simple and straight
-
forward method
of minimizing Boolean expressions


With the Karnaugh map Boolean expressions having up to four and
even six variables can be simplified.


Karnaugh map provides a pictorial method of grouping together
expressions with common factors and therefore eliminating
unwanted variables.


Karnaugh map can also be described as a truth table.

Karnaugh Maps


Minterm: (Standard product or canonic product term) such as AB’CD
or A’BCD’ etc. where each variable used once and once only.


Maxterm: (Standard sum or canonical sum term) such as
(A+B’+C+D) or (A’+B+C+D’) where each variable used once and
once only


Sum of products: (Minterm canonic form or canonic sum function


f(A,B,C,D)=AB’CD+A’BCD’+A’BC’D


Product of sums: (Maxterm canonic form or canonic product function


f(A,B,C,D)=(A+B’+C+D) (A’+B+C+D’)(A’+B+C’+D)


Adjacent Cells: If two occupied cells of a Karnaugh are adjacent,
horizontally, vertically (but not diagonally) then one variable is
redundant. Adjacent cells differ in the value of only one variable.


(Rule of adjacency
-

can knock off one variable as A+A’=1)





Karnaugh Maps


Combining all adjacent 1’s more than once doesn’t matter unless no
1 is left out, as A + A = A and A.A = A


Physical, Logical adjacency



Karnaugh Maps


The correspondence between the Karnaugh map and the truth table



(two variable)


The values inside the squares are copied from the output column of the truth
table, therefore there is one square in the map for every row in the truth table



Karnaugh Maps


Consider the following map. The function plotted is:


Z = f(A,B) = A B’+ AB


Referring to the map the two 1’s are grouped together. The variable B has
its true and false form within the group. Eliminate B leaving only A which
only has its true form







Using algebric simplication:


Z = A


+ AB

Z = A(


+ B)

Z = A


Karnaugh Maps


Consider the expression z = f(A,B) = A’B’+AB’+A’B plotted on
Karnaugh map:


The first group labeled I, consists of two 1s which correspond to A =
0, B = 0 and A = 1, B = 0. Put in another way, all squares in this
example that correspond to the area of the map where B = 0
contains 1s, independent of the value of A. So when B = 0 the
output is 1. The expression of the output will contain the term



For group labeled II corresponds to the area of the map where A = 0. The
group can therefore be defined as . This implies that when A = 0 the output
is 1. The output is therefore 1 whenever B = 0 and A = 0


Hence the simplified answer is Z = A’ + B’


Karnaugh Maps


Given the truth table


The Boolean algebraic expression is

m = a'bc + ab'c + abc' + abc.

Minimization is done as follows.

m = a'bc + abc + ab'c + abc + abc' + abc


= (a' + a)bc + a(b' + b)c + ab(c' + c)


= bc + ac + ab

bc + ac + ab

Karnaugh Maps


The Karnaugh map for 4 variables

q = a'bc'd + a'bcd + abc'd' + abc'd + abcd + abcd' + ab'cd + ab'cd'

RULE: Minimization is achieved by drawing the smallest possible
number of circles, each containing the largest possible number of 1s.


q = bd + ac + ab

This expression requires 3 2
-
input
and

gates and 1 3
-
input
or

gate.

Karnaugh Maps


Imlpicant:


Each of the terms i.e product terms that are combined to become sum of products
later on are called implicants


Prime Imlpicant:


Largest possible group of values. For that group we can not find larger group


An implicant can get submerged in to a prime implicant.


Essential Prime Imlpicant:


At least one 1 or cell, which not been covered in any other group should be covered


Non Essential Prime Imlpicant:


One way of combining 1s which is not covered otherwise as a an essential prime
implicant


Gordon E. Moore


Each year computer chips become more
powerful yet cheaper than the year before.
Gordon Moore, one of the early integrated circuit
pioneers and founders of Intel once said,



"If the auto industry advanced as rapidly as the
semiconductor industry, a Rolls Royce would get
a half a million miles per gallon, and it would be
cheaper to throw it away than to park it.”