unit 1 - boolean algebra

8/2/2019 Unit 1 - Boolean Algebra

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Combinational vs.Sequential Circuits

Acombinationalcircuithas inputs, outputs andan internal logic circuit

Constitutes a mappingfrom the inputs to theoutputs

Asequentialcircuit hasinputs, outputs, aninternal logic circuit and

binary cellsas memory

Abinary cellis a memorydevice with two possible states.


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Boolean Algebra

Variables in a Booleanalgebra take on valuesthat are not numbers

but logical values:TRUE or FALSE

x= TRUE, y = FALSE George Boole1815 - 1864

It is not of the essence of mathematics to be conversant with

the ideas of number and quantity.

~ An Investigation into the Laws of Thought, on Which are

Founded the Mathematical Theories of Logic and Probabilities


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LOW

HIGH

Logical Variables

Logical variables takeonly two values:

1.0

3.0

0.4

0.2

Voltage

Level[V]

In actual computers,

these are representedby voltage levels

TRUE 1 HIGHFALSE 0 LOW


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Logical Operators

Logical operatorsmathematicallyassociate one or more

(logical) input valueswith a single (logical)output value

Logic gates are thephysical devices that

perform logicaloperations (forinstance, in circuits)

Basic Logical Operators

One Input

Two (or More) Inputs

NOT(negation)

OR

(disjunction)

AND(conjunction)


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Switching Circuits

Boolean algebra was firstapplied to switching circuits

ORgate

ANDgate

Light

Bulb

Battery

LightBulb

Battery

xy

xy

x y

y

x

In the input, use a '1' to represent a closed switch, a'0' for an open switch; in the output, use a '1' for a lit

bulb, a '0' for an unlit bulb


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Truth Tables

Since there are only two possible values (0 or 1) for

any logical variable, we can define each of thelogical operations by the output they produce givenall of the possible inputs

x

0 11 0

NOT

xx

NOTgate(inverter)

xor

(Logic) xor

x

F TT F

x


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Truth Tables

x y xy

0 0 00 1 0

1 0 0

1 1 1

AND

xy

*x y

ANDgate

xyor

x y(Logic)or


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Truth Tables

x y x + y

0 0 00 1 1

1 0 1

1 1 1

OR

x

yx + y

ORgate

x y(Logic)or


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Order of Operations

The logical operatorshave an order ofprecedence:

1. NOT

2. AND

3. OR

F = x + yz

G x + y z

H = x + yz

J x + yzAs usual brackets canalways be used tochange the order ofprecedence

BooleanExpressions


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Boolean Postulates

The truth tables for each of the basic logicaloperations can also be stated as postulatesfor anew kind of algebra a Boolean algebra

0 0 00*0 0 P2a P2b

1 1 11*1 1 P3a P3b

0 1 11*0 0 P4a P4b

1 0 0 1 P5a P5b

1 if 0 0 if 1x x x x P1a P1b

Boolean

algebra is

closed overthe values

{0, 1}


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Boolean Algebra

This new algebra of logical operators has severalproperties in common with the more familiaralgebra of addition and multiplication

Commutative Property

Associative Property

The order in which Boolean

variables areANDedorOReddoesnt matter to the result

The manner in which ANDedorORedBoolean variablesare groupeddoesnt matter to

the result

* *

x y

x y y x

y x

* * * *

x y

x

z x

y z

y

z

z

x y


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Boolean Algebra

Boolean algebra has one more property that is notfamiliar from the algebra of addition and multiplication

Distributive Property I

* * *x y z x y x z

Distributive Property II

* *x y z x y x z


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Relation to Set Theory

The logical operators are related to set theory

NOT

OR

AND

x

x + y

*x y

complement

union

intersection

x

x y

x y

x

x yx yx y


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In terms of set theory, this corresponds to:

0 0 00*0 0 P2a P2b

There is no intersection between oneempty set and another empty set.

The union of one empty set and

another empty set is still empty.

Note: Since a 0indicates a Boolean value FALSE,the set ofTRUE values with which it is associated is

empty. Hence, a 0 corresponds to the empty set.


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0 1 11*0 0 P4a P4b

There is no intersection between theentire universe and an empty set.

The union of the entire universe and

an empty set is still the universe.

U U

U

Note: Since a 1indicates TRUE, the set ofTRUEvalues with which it is associated is the entire universe

of possibilities. Hence, a 1 corresponds to the universe.


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Boolean Theorems


*0 0x T9a

There is no intersection between a

setx and an empty set.x

In terms of truth tables, this corresponds to:

x x *00 0 by P2a

1 0 by P4a


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Boolean Theorems

0x x T9b

The union of a setx and an empty

set is still the setx.x x



x x +00 0 by P2b

1 1 by P4b

0 is the identityelementwithrespect to OR


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Boolean Theorems

1 1x T10b

The union of a setx and the universeis still the universe.

x U U



x x +10 1 by P4b

1 1 by P3b


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Boolean Theorems

* x x x xx x T11a T11b

0 1* x xx x T12a T12b

Idempotency Theorems

Inverse Elements

x x xx T13a T13b

Double Negation

Both AND and OR are idempotent, meaning that ANDing orORing a Boolean variable with itself once has the same effectas ANDing or ORing the variable with itself many times.

Every Boolean variable has an inverse element given in termsof the NOT operator.


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Absorption Theorems

Absorption theoremsallow the simplification ofcircuits by reducing the

number of gates.x xy x T14aProve

x y

x xyx xy

(identity element for )*1 *x xy x x y by T10a AND

(distribution)* 1x y by P8a*1x by T10b

(identity element for )x by T10a AND


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Proof Using Truth Tables

Boolean proofs may be completed using the theoremsor, equivalently, using truth tables

x

0

01

1

y

0

10

1

xy

0

10

0

x + xy

0

11

1

x + y

0

11

1

x x y x y T14dProve

x1

10

0

Permutationsare listed in

the order ofthe binarynumbers

from 0 to 3


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F0

0

0

0

0

0

0

0

0

F1

1

0

0

0

0

0

0

0

F2

0

1

0

0

0

0

0

0

Proof Using Truth Tables

ForNvariables, there are 2Npermutations of 0 and 1

There are therefore

possible Boolean functions

forNvariables

y

0

0

1

1

0

0

1

1

z

0

1

0

1

0

1

0

1

For instance, for three variables,N= 3:

There are 23= 8 permutations.

Permutations are listed in the order of the

binary numbers from 0to 7.

There are 28= 256 Boolean functions.

x

0

0

0

0

1

1

1

1

22N

F3

0

0

1

0

0

0

0

0

F4

0

0

0

1

0

0

0

0

F5

0

0

0

0

1

0

0

0


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Derived Boolean Operators -NAND

x y xy (xy)0 0 0 1

0 1 0 1

1 0 0 1

1 1 1 0

NAND

xy

NANDgate

xy

or NANDx y

or

x y


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Derived Boolean Operators XOR (Exclusive OR)

x y

0 0 0

0 1 1

1 0 1

1 1 0

XOR

xy

XORgate

x y

or

XORx y

x y


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Derived Boolean Operators XNOR (Exclusive NOR or Equivalence)

x y

0 0 0 1

0 1 1 0

1 0 1 0

1 1 0 1

XNOR

xy

XNORgate

or

XNORx y

x y x y

x y

x yor

unit 1 - boolean algebra

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