dld lecture 2

7/29/2019 DLD Lecture 2

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Amina Asghar

Boolean Algebra and Logic Gates


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Binary (Boolean) Logic

y Deals with binary variables and binary logic functions

y Has two discrete values

y 0False,Open

y1True,Close

y Three basic logical operations

y AND (.); OR (+); NOT ()


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Logic Gates and Truth Table


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Boolean Algebra

y Branch of Algebra used for describing and designing two

valued state variables

y Introduced by GeorgeBoole in 19th centaury

yShannon used it to design switching circuits (1938)


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Boolean Algebra Postulatesy An algebraic structure defined by a set ofelements,B,

together with two binary operators + and . that satisfythe]following postulates:

y Postulate1:

y Closure with respect to both +

y Closure with respect to both .

y Postulate 2:

y An identity element with respect to +, designated by 0 ( x+0

= 0+x = x)y An identity element with respect to . designated by 1 ( x.1=1.x = x)


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y Postulate3:y Commutative with respect to + ( x+y = y+x)y Commutative with respect to . ( x.y = y.x)

y Postulate 4:y

. is distributive over + ( x.(y+z) = (x.y)+(y.z) )y + is distributive over . ( x+(y.z) = (x+y).(y+z) )

y Postulate 5:y For each element a ofB, thereexist an element a such that

y a + a = 1

y a. a = 0

y Postulate 6:y Thereexists at least two elements a, b in B, such that a b


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Boolean Algebra - Theorems

Theorems help us out in manipulating Boolean expression

They must be proven from other postulates or already proven theorems

Theorems help us out in manipulating Boolean expression

They must be proven from other postulates or already proven theorems


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Duality Principle of Boolean Algebra

y Duality Principle says that every algebraic expression

deducible from the postulates ofBoolean algebra remains

valid if theoperators and identity of elements are

interchanged.

y In two valued Boolean algebra, the identity elements and

elements of set B are same: 1 & 0

y If dual of algebraic expression is required, we simply

interchange OR and AND operators.


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Lets prove the following theorems using

postulates/ proven theorems

y Theorem 1(a): x+x=x

y Theorem 1(b): x.x=x by duality

y Theorem 2(a): x+1=1

y Theorem 2(b): x.0=0 by dualityy Theorem 3: (x)=x

y Theorem 6(a): x+xy=x

y Theoem 6(b)= x(x+y)=x by duality

The theorems ofBoolean algebra can also be shown true by means of truth

table


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Operator Precedence for Boolean

Expression

y Parentheses

y NOT

y AND

y OR


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Boolean Functions

y A Boolean function is an expression formed with binary

variables, binary operators OR and AND, unary operator

NOT, parentheses and equal sign.

y

Examplesy F

1= xyz

y F2= x + yz

y F3= xyz + xyz + xy


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y A Boolean function may also be represented by truth table

Same functions


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Representation of Boolean function by

logical diagram


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Implementation of which function

requires less gates F3 or F4??


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Algebraic Manipulationsy Literal primed or unprimed variable

y When Boolean function is implemented with logic gates,eachliteral in the function is designated as input to gate and each termis implemented with a gate.

y Complex Boolean function --- large number of gates

y To get simpler circuits, one must know hoe to manipulateBoolean functions to obtain equal and simpler expression.y Literal minimization and term minimizationy We can do the literal minimization by applying algebraic

manipulationsby employing postulates and basic theorems.

y Term minimization will be discussed later.


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Examples

y Simplify the following

y x + xy

y x (x+ y)

y xyz + xyz + xyy xy + xz + yz

yy ((x+yx+y) () (x+zx+z) () (y+zy+z))


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Complement of a Function

y F is a complement ofF and is obtained by an interchange of

0s for 1s and 1s for 0s.

y Complement of function can be obtained by De Morgans

theorem


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Example

y Find the complement of the following functions by applying

De Morgans theorem as many times as necessary.

y F1= xyz + xyz

y F2= x (yz+ yz)


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Example

y Find the complement of following functions by taking their

dual and complementing each literal.

y F1= xyz + xyz

y F2= x (yz+ yz)


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Canonical forms of Expressiony We can writeexpressions in many ways, but some ways are more useful than

others

y A sum of products (SOP) expression contains:

y Only OR (sum) operations at the outermost level

y

Each term that is summed must be a product of literalsy The advantage is that any sum of products expression can be implemented using

a two-level circuit

y literals and their complements at the 0th level

y AND gates at the first level

y a single OR gate at the second level


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Minterms

Primed variable corresponds

to 0

and unprimed correspond

to 1

Primed variable corresponds

to 0

and unprimed correspond

to 1


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Sum of Minterms form


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The dual idea


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Maxtems

Primed variable

corresponds to 1 andunprimed corresponds

to 0

Primed variable

corresponds to 1 andunprimed corresponds

to 0


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Product of Maxterms form


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Minterms and Maxterms are related


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Converting between Canonical forms


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Sum of Minterms

y We investigated that

y For n variables there are 2n minterms

y Any boolean function can be described in terms of sum of

minterms

y The function can beeither 1 or 0 for each minterm, and

since there 2n minterms, therefore the possible functions that

can be formed with n variables are 22n


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Expressing the Boolean function as

sum of minterms

y Expand theexpression into sum of minterms

y Each term is inspected to check that it contains all the variables

y If the term misses any variable then it is ANDed with an expression such

as x+x,where x is the missing variable


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Example


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Expressing the function in terms of

product of Maxterms

y Each 22n functions of n variables can also beexpressed as a

product of maxterms

y To express a function in terms of product of maxterms

y

It must first rewritten as OR terms by using distributive lawx+yz=(x+y)(x+z)

y Then the missing variable in each term is ORed with xx


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Example


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Standard forms of Expression

y In standard forms, terms may have one, two or any number

of literals

y Sum of products is a Boolean expression containing AND

terms of one or more literals each.The sum denotes the

ORing of these terms.

y Product of sums is a Boolean expression containing OR

terms of one or more literals each.T

he product denotes theANDing of these terms


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Non standard form

y Sometimes expression is neither sum of products or product

of sums.This is non standard form.

y It can be changed into standard form by using distributive

law .


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Digital Logic Gates


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0


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dld lecture 2

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