Chapter 2 Boolean Algebra & Logic Gates

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Princess Sumaya Univ.Computer Engineering Dept.

Chapter 2:


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Princess Sumaya University 4241Digital Logic Design Computer Engineering Dept.

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Basic Definitions

Binary Operators

AND

z= x y= x y z=1 if x=1 AND y=1

OR

z= x+ y z=1 if x=1 OR y=1

NOT

z= x=x z=1 if x=0

Boolean Algebra

Binary Variables: only 0 and 1 values

Algebraic Manipulation


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Boolean Algebra Postulates

Commutative Law

x y= y x x+ y= y+ x

Identity Element

x 1 = x x+ 0 = x

Complement

xx= 0 x+x= 1


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Boolean Algebra Theorems

Duality

The dualof a Boolean algebraic expression is obtained

by interchanging the ANDand the ORoperators and

replacing the 1s by 0s and the 0s by 1s.

x ( y+ z) = ( x y) + ( x z)

x+ ( y z) = ( x+ y) ( x+ z)

Theorem 1

x x= x x+ x= x

Theorem 2

x 0 = 0 x+ 1 = 1

App l ied to a validequat ion prod uces

a validequat ion


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Boolean Algebra Theorems

Theorem 3: Involution

(x)= x ( x) = x

Theorem 4: Associative & Distr ibutive

( x y) z= x ( y z) ( x+ y) + z= x+ ( y+ z)

x ( y+ z) = ( x y) + ( x z)

x+ ( y z) = ( x+ y) ( x+ z)

Theorem 5: DeMorgan

( x y)=x +y ( x+ y)=x y ( x y)= x + y ( x+ y)= x y

Theorem 6: Absorption

x ( x+ y) = x x+ ( x y) = x


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Operator Precedence

Parentheses

(. . . ) (. . .)

NOT

x+ y

AND

x+ xy

OR

])([ xwzyx

])([

)(

)()(

)(

xwzyx

xwzy

xwz

xw

xw


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DeMorgans Theorem

)]([ edcba

)]([ edcba

))(( edcba

))(( edcba

))(( edcba

)( edcba


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Boolean Functions

Boolean Expression

Example: F= x+yz

Truth Table

All possible combinationsof input variables

Logic Circuit

x y z F0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1

x

yz

F


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Algebraic Manipulation

Literal:

A single variable within a term that may be complemented

or not.

Use Boolean Algebra to simplify Boolean functions

to produce simpler circuits

Example: Simplify to a minimum number of literals

F= x+x y ( 3Literals)

= x+ (x y )

= ( x+x) ( x+ y)

= ( 1 ) ( x+ y) = x+ y ( 2Literals)

Distr ibut iv e law(+ over )


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Complement of a Function

DeMorgans Theorm

Duality & Literal Complement

CBAF

CBAF

CBAF

CBAF

CBAF

CBAF


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Canonical Forms

Minterm

Product (ANDfunction)

Contains all variables

Evaluates to 1 for a

specific combination

Example

A= 0 A B C

B= 0 (0) (0) (0)

C= 01 1 1= 1

A B C Minterm

0 0 0 0 m0

1 0 0 1 m1

2 0 1 0 m2

3 0 1 1 m3

4 1 0 0 m4

5 1 0 1 m5

6 1 1 0 m6

7 1 1 1 m7

CBA

CBA

CBA

CBA

CBA

CBA

CBA

CBA


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Canonical Forms

Maxterm

Sum (ORfunction)

Contains all variables

Evaluates to 0 for a

specific combination

Example

A= 1 A B C

B= 1 (1) + (1) + (1)

C= 10 + 0 + 0= 0

A B C Maxterm

0 0 0 0 M0

1 0 0 1 M1

2 0 1 0 M2

3 0 1 1 M3

4 1 0 0 M4

5 1 0 1 M5

6 1 1 0 M6

7 1 1 1 M7

CBA

CBA

CBA

CBA

CBA

CBA

CBA

CBA


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Canonical Forms

Truth Table toBoolean Function

A B C F

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 11 1 0 0

1 1 1 1

CBAF CBA CBA ABC


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Standard Forms

Sum of Products (SOP)

ABCCBACBACBAF

AC

BBAC

)(

CB

AACB

)(

BA

BA

CCBA

)1(

)(

)()()( BBACCCBAAACBF

ACBACBF


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Standard Forms

Product of Sums (POS)

)( AACB

)( BBCA

)( CCBA

)()()( AACBCCBABBCAF

CBBACAF

CABBCACBACBAF

))()(( CBBACAF


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Two-Level Implementations

Sum of Products (SOP)

Product of Sums (POS)

ACBACBF

))()(( CBBACAF

B

C

FBA

AC

AC

FBA

B

C


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Logic Operators

AND

NAND (NotAND) x y NAND0 0 1

0 1 11 0 1

1 1 0

x y AND

0 0 0

0 1 0

1 0 0

1 1 1

x

yx

y

xy

x y


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Logic Operators

OR

NOR (NotOR)

x y OR

0 0 0

0 1 1

1 0 1

1 1 1

x y NOR

0 0 1

0 1 01 0 0

1 1 0

xy

x+ y

x

y x+ y


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Logic Operators

XOR (Exclusive-OR)

XNOR (Exclusive-NOR)

(Equivalence)

x y XOR

0 0 0

0 1 1

1 0 1

1 1 0

x y XNOR

0 0 1

0 1 0

1 0 0

1 1 1x

y

x y

x yxy + xy

x

yx y

xy + xy


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Logic Operators

NOT (Inverter)

Buffer

x NOT

0 1

1 0

x Buffer

0 0

1 1

x x

x x


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Multiple Input Gates


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DeMorgans Theorem on Gates

AND Gate

F= x y F= (x y) F= x+ y

OR Gate

F= x+ y F= (x+ y) F= x y

Change the Shape and bubble all lines

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Chapter 2 Boolean Algebra & Logic Gates