ES ZC261-L2

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BITS PilaniPilani Campus

BITS Pilani

presentationRekha.A

Faculty

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BITS Pilani, Pilani Campus

Boolean Algebra like any other deductive mathematical system, may bedefined with a set of elements, set of operators and a number of Postulates.

Boolean algebra differs in a major way from the ordinary algebra in thatBoolean constants and variables are allowed to have only 2 possible values0 or 1. Thus the Boolean 0 or 1 do not represent the actual numbers but

instead represent the state of a voltage variable or what is called as logiclevels.

In Boolean algebra there are 3 basic operations: OR, AND, NOT. Thesebasic operations are called as Logic operations. Digital circuits called LogicGates can be constructed in such a way that the circuit o/p is the result of abasic logic operations performed on the inputs.

ESZC261:Digital Electronics and Microprocessor2

Logic gates and Boolean Algebra.

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The most common laws used to formulate various algebraic structures are

as follows:

Associative Law: (x*y)*z = x*(y*z) for all x,y,z S

Commutative Law: A binary operator * on a set S is said to be commutative

, whenever

x*y = y*x for all x.y S



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Identity Element: A set S is said to have an identity element

with respect to a binary operation * on S if there is an element

e S with the property thate * x = x * e = x, for every x S.

Distributive Law : If + and . Are the two binary operators on

a set S, . is said to be distributive over . Whenever

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

ESZC261:Digital Electronics and Microprocessor 4


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Postulates and theorems of Boolean algebra

x + 0 = x

x.1=x x.x=x

x.x=0

x+0=x

x+1=1

x+x=x x+x=1

(x+y)= xy and (xy)=x+y De morgans Theorem


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Verification of Demorgans theorem

1. X Y X+Y (X+Y) X Y XY

0 0 0 1 1 1 1

0 1 1 0 1 0 0

1 0 1 0 0 1 01 1 1 0 0 0 0

2. X Y XY (XY) X Y X+Y

0 0 0 1 1 1 10 1 0 1 1 0 1

1 0 0 1 0 1 1

1 1 1 0 0 0 0


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Logic Gates :

Building Blocks of Digital Systems.

Basic gates

AND, OR, NOT

Universal Gates

NAND, NOR

Derived Gates

X-OR, X-NOR

ESZC261 Digital Electronics and Microprocessors

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OR Operation:

Let A and B represent two independent logic variables. When A and B

are combined using OR operation , The result x can be expressed as X = A+ B

IC 7432



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Truth table of OR Gate.

Input Out put

A B C

0 0 0

0 1 1

1 0 1

1 1 1



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Use of OR gate in alarm system:


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AND operation:

If the two logic variables A and B are combined using AND operation , the result

X can be expressed as X = A.B

AND Gate:

An AND gate is a logic circuit with two or more than two inputs and one output.

The output of an AND gate is logic1 only when all of its inputs are logic1.In

all other cases the output is logic 0.



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BITS Pilani, Pilani CampusESZC261:Digital Electronics and Microprocessor 12

IC 7408

. Logic gates and Boolean Algebra

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Truth table of AND Gate.

Input Out put

A B C

0 0 0

0 1 0

1 0 0

1 1 1



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Truth Table of 3 Input AND gate

Input Out put

A B C Y

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1



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NOT Operation:

If a Variable A is subjected to the

NOT operation the

result X can be expressed as

X = A

IC 7404

Input Out put

A B

0 1

1 0


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NAND GATE:

An AND gate followed by a NOT circuit make it as a NAND gate. NAND

gate is a logic circuit with two or more than two inputs and one output. The

output of NAND gate is logic0 only when all of its inputs are logic1.In all

other cases the output is logic 1


IC 7400

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Truth table of NAND Gate.

Input Out put

A B C

0 0 1

0 1 1

1 0 1

1 1 0


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NOR Operation:

An OR gate followed by a NOT circuit make it as a NOR gate. An NOR gate is

a logic circuit with two or more than two inputs and one output. The output of

NOR gate is logic 1 only when all of its inputs are logic1.In all other cases

the output is logic0.


IC 7402

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Truth table of NOR Gate.

Input Out put

A B C

0 0 1

0 1 0

1 0 0

1 1 0


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XOR GATE:

Exclusive OR gate has high output only when an odd number of inputs are

high.


IC 7486

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Truth table of EX-OR Gate.

Input Out put

A B C

0 0 0

0 1 1

1 0 1

1 1 0


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Describing logic circuits Algebraically

Any logic circuit can be described using the Boolean operations.

For example consider the following circuit


This Circuit has 3 inputs A,B,C and a single output Y.

The Expression for Y = A.B + C

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Example 2

For the circuit shown write the algebraic expression


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Boolean Functions

A Boolean function is described by an algebraic expression consisting of

binary variables, the constants 0 & 1 and the logic operation symbols

For the function F1 = x+ yz . Write the function table . write the Gate

ImplementationX Y Z Y X+YZ

0 0 0 1 0

0 0 1 1 1

0 1 0 0 0

0 1 1 0 0

1 0 0 1 1

1 0 1 1 1

1 1 0 0 1

1 1 1 0 1


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Gate Implementation of the function F= X+YZ

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Find the complement of the function F = xyz+xyz

Solution:

F = (xyz+xyz)

= (xyz)(xyz)

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

Find the complement of the function F=x(yz+yz)

F= {x(yz+yz)}

= x+(yz+yz)

= x+ (yz)(yz)

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


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MINTERMS(Sum of Products)

A binary variable may appear either in its normal form(x) or incomplement form(x)

The two binary variables x and y combined with an AND operation.

The four Possible combinations are xy, xy, xy, xy.

xy, xy, xy, xy are called Minterms.

n variables can be combined to form 2n minterms

x Y term

0 0 XY m00 1 XY m11 0 XY m21 1 XY m3

f= xy+xy = mo+m2


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MAXTERMS(Product of Sum)

The two binary variables x and y combined with an OR operation.

The four Possible combinations are x+y, x+y, x+y, x=y.

x+y, x+y, x+y, x+y are called Maxterms.

n variables can be combined to form 2n maxterms

x Y term

0 0 X+Y M00 1 X+Y M11 0 X+Y M2

1 1 X+Y M3

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

= M3 . M1. M0


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Convert the following equation into standard SOP form

y = AB+AC+BC

Solution:

y= AB(C+C) + AC(B+B) + BC(A+A)

= ABC+ABC+ABC+ABC+ABC+ABC

= ABC+ABC+ABC+ABC

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Convert the equation into standard POS form

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

Solution:Y=(A+B+CC)(A+BB+C)(AA+B+C)

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

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

ES ZC261-L2

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