Boolean Algebra and Logic gates · easier to fabricate and are the basic gates used in all IC...

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Module 3 : Boolean Algebra and Logic gates 1

Module-3

Boolean Algebra and Logic gates 3.1. Motivation:

Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital logic and Boolean algebra.

3.2. Syllabus:

Module Contents Duration Self-Study

3.1

Introduction, NAND and NOR operations, Exclusive –

OR and Exclusive –NOR operations, 1Lectures 1 Hour

3.2 Boolean Algebra Theorems and Properties 1 Lectures

1 Hour

3.3 Standard SOP and POS form, 1 Lectures

1 Hour

3.4 3.5

Reduction of Boolean functions using Algebraic method, K -map method (2,3,4 Variable)

2 Lectures 4 Hours

3.6 Variable entered Maps,

1 Lecture 1 Hour

3.7 Quine Mc Cluskey, 1 Lecture

2 Hour

3.8 Mixed Logic Combinational Circuits and multiple

output function 1 Lecture

2 Hour

3.9

3.10

Basic Digital Circuits: NOT, AND, OR, NAND, NOR,

EX-OR, EX-NOR Gates. 2 Lecture

2 Hours

3.3. Weightage in university Examination: -

3.4. Learning Objective/ outcome :

3.4.1 Learning Objective: In this module student will try Describe Various Number system and Boolean algebra. 3.4.2 Learning Outcome: At the end student will be able to Minimize the Boolean expression using Boolean algebra and design it using logic gates

3.5. Theoretical Background: One has to have basic knowledge of digital electronics, to describe and understand the

analysis and design of basic gates.

3.6. Abbreviations:

POS: Product Of sum

SOP: Sum OF Product

BCD: Binary Coded Decimal

ASCII: American Standard Code for Information Interchange

LSB: Least Significant Bit

MSB: Most Significant Bit

LSD: Least Significant Digit

MSD: Most Significant Digit

3.7. Key Definitions:

Karnaugh maps: A mapping technique used to minimize logical expression

Minterm: A logical term consisting of all the literals in the ANDed form.

Maxterm: A logical term consisting of all the literals in the ORed form in the logic

function

Canonical Form: A term used to describe a Boolean function that is written either as

a sum of minterms, or as a product of maxterms. For example, using three variables

A, B, and C, the equation f (A, B, C) = A(B ~ C) + ABC

Product of Sum (POS): A logical expression in the form of ORed terms ANDed

together

Sum of Product (SOP): A logical expression in the form of ANDed terms ORed

together

Literal: A Literal is a Boolean Variables or its complement

Combinational logic: The logic in which the outputs at any instant of time are dependent only on the inputs present at that time

Standard POS: If each term in POS form contains all the literals then the POS form is known as standard or canonical POS form

Standard SOP: If each term in SOP form contains all the literals then the SOP form is

known as standard or canonical SOP form.

Don’t care: A minterm/maxterm in a logic function may or may not be included.

Essential Prime implicants: A tern in SOP form in a logic function that must be present in the minimal expression.

3.8 Introduction

3.8.1 Basic Logic Gates

The term ‘Gate’ is used to describe the members of a set of basic electronic

components which, when combined with each other, are able to perform complex

logical and arithmetic operations. 'Gates' are the physical realization of the simple

Boolean expressions.

AND Gate:

The AND gate is a circuit which gives a high output (logic 1) if all its inputs are

high. Otherwise the output will be low (logic 0).

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Fig 3.1

A B AB

0 0 0

0 1 0

1 0 0

1 1 1

Table 3.1

OR Gate:

The OR gate is a circuit which

gives a high output (logic 1) if

any one of the inputs is high.

Otherwise the output will be

low

Fig 3.2

A B A+B

0 0 0

0 1 1

1 0 1

1 1 1

Table 3.2

NOT Gate:

Fig 3.3

The NOT gate is a circuit

which produces at its output

the negated (inverted) version

of its input logic. The circuit is

also known as an inverter.

A NOT

0 1

1 0

NAND Gate: Table 3.3

The NAND gate is a NOT-AND

circuit which is equivalent to an

AND circuit followed by a NOT

circuit. The output of the NAND

gate is high if any of its inputs is

low.

Fig 3.4

A B NOT(AB)

0 0 1

0 1 1

1 0 1

1 1 0

Table 3. 4

NOR Gate:

The NOR gate is a NOT-OR

circuit which is equivalent to an

OR circuit followed by a NOT

circuit. The output of the NOR

gate is low if any of its inputs is

high.

A B NOT

(A+B)

0 0 1

0 1 0

1 0 0

1 1 0

Fig 3.5 Table 3.5

Exclusive-OR Gate (EOR):

Fig 3.6

The Exclusive-OR (EOR) gate

is a circuit which gives a high

output if either of its two

inputs is high, but not both. A

encircled plus sign ( ) is used

to indicate the EOR operation.

A B A

B

0 0 0

0 1 1

0 1

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1

1 1 0

Table 3.6

Universal Gates:

A universal gate is a gate which can implement any Boolean function without

need to use any other gate type. The NAND and NOR gates are universal gates.

In practice, this is advantageous since NAND and NOR gates are economical and

easier to fabricate and are the basic gates used in all IC digital logic families. In

fact, an AND gate is typically implemented as a NAND gate followed by an

inverter not the other way around!! Likewise, an OR gate is typically

implemented as a NOR gate followed by an inverter not the other way around!!

NAND Gate is a Universal Gate:

To prove that any Boolean function can be implemented using only NAND gates,

we will show that the AND, OR, and NOT operations can be performed using

only these gates.

Implementing an Inverter Using only NAND Gate

The figure shows two ways in which a NAND gate can be used as an inverter

(NOT gate).

1. All NAND input pins connect to the input signal A gives an output A’.

Fig 3.7

2. One NAND input pin is connected to the input signal A while all other input

pins are connected to logic 1. The output will be A’.

Fig 3.8

3. Implementing AND Using only NAND Gates

An AND gate can be replaced by NAND gates as shown in the figure (The AND

is replaced by a NAND gate with its output complemented by a NAND gate

inverter).

Fig 3.9

4. Implementing OR Using only NAND Gates

An OR gate can be replaced by NAND gates as shown in the figure (The OR gate

is replaced by a NAND gate with all its inputs complemented by NAND gate

inverters).

Fig 3.10

Thus, the NAND gate is a universal gate since it can implement the AND, OR

and NOT functions.

NOR Gate is a Universal Gate:

To prove that any Boolean function can be implemented using only NOR gates,

we will show that the AND, OR, and NOT operations can be performed using

only these gates.

1. Implementing an Inverter Using only NOR Gate

The figure shows two ways in which a NOR gate can be used as an inverter (NOT

gate).

1. All NOR input pins connect to the input signal A gives an output A’.

Fig 3.11

2. One NOR input pin is connected to the input signal A while all other input pins are

connected to logic 0. The output will be A’.

Fig 3.12

3. Implementing OR Using only NOR Gates

An OR gate can be replaced by NOR gates as shown in the figure (The OR is

replaced by a NOR gate with its output complemented by a NOR gate inverter)

Fig 3.13

4. Implementing AND Using only NOR Gates

An AND gate can be replaced by NOR gates as shown in the figure (The AND

gate is replaced by a NOR gate with all its inputs complemented by NOR gate

inverters)

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Fig 3.14

Thus, the NOR gate is a universal gate since it can implement the AND, OR and

NOT functions.

Boolean algebra consists of binary variables and logical operations. Variables:

letters, such as A, B, C, d, W, x, y, z, etc. having two distinct possible values 1 and

0.

Logical operations: AND, OR and NOT

AND: represented by a dot or the absence of an operator. x . y=z or xy=z, this

reads: "x AND y is equal to z”. It means that z=1 if and only if [x=1 and y=1;

otherwise z=0]

OR: represented by a plus sign. x + y=z , is read "x OR y equals to z". it means

that z=1 if either [x=1 or y=1; if both are zero then z=0]

NOT: represented by a prime or a bar. x'=z , reads "not x equals to z". it means

that if x=1 then z=0 or if x=0 then z=1.

These Logical Operations are digital circuits constructed from diodes, transistors,

and resistors connected in such a way that the circuit output is the result of a

basic logic operation performed on the inputs.

B) Truth Table is a table of all possible combinations of the variables showing the

relation between the values that variables may take and the result of the

operation. In another way one can say a truth table is a means for describing how

a logic circuit's output depends on the logic levels present at the circuit's inputs.

The following is truth table example of a two input logic circuit,

Input

1

Input

2 Output

X Y Z = X ?

Y

0 0 0

0 1 0

1 0 0

1 1 1

Table 3.7

Since the output Z is 1 only if X and Y are 1, therefore, the "?" is an AND gate.

C) Boolean Identities are used to designate the function, dictates the

interconnection of gates in the logic circuit diagram. By manipulating Boolean

expression according to Boolean algebraic rules. It is possible to obtain a simpler

expression with the same function. such manipulation will reduce the number of

gates the number of inputs to the gates.

Using the relations defined in the AND, OR and NOT operation, a number of

identities are stated.

Group A Group B

1 X + 0 = X X . 1 = X

2 X + 1 = 1 X . 0 = 0

3 X + X = X X . X = X

4 X + X' = 1 X . X' = 0

5 (X')' = X

6 Commutative X+Y = Y+X XY = YX

7 Associative X+(Y+Z)=

(X+Y)+Z

X(YZ) =

(XY)Z

8 Distributive X (Y + Z) =

XY + X Z

X + YZ =

(X + Y)(X +

Z)

9 DeMorgan's

Theorem (X+Y)'=X'Y' (XY)'=X'+Y'

Table 3.8

D) Duality Principle: Every algebraic expression deducible from the identities of

Boolean algebra remains valid if the operators and the identitiy elements are

interchanges.

Dual of an expression: Interchange OR and AND operators and replace 1's by 0's

and 0's by 1's

An Equation in one column can be obtained from the corresponding equation in

the other column by taking the dual of the expressions.

3.8.3. Standard SOP and POS form, Reduction of Boolean functions using Algebraic

method, K -map method (2, 3, 4 Variable).

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Since there are a finite number of Boolean functions of n input variables, yet an infinite

number of possible logic expressions you can construct with those n input values, clearly

there are an infinite number of logic expressions that are equivalent (i.e., they produce the

same result given the same inputs). To help eliminate possible confusion, logic designers

generally specify a Boolean function using a canonical, or standardized, form. For any given

Boolean function there exists a unique canonical form. This eliminates some confusion

when dealing with Boolean functions.

Actually, there are several different canonical forms. We will discuss only two here and

employ only the first of the two. The first is the so-called sum of minterms and the second is

the product of maxterms. Using the duality principle, it is very easy to convert between these

two. A term is a variable or a product (logical AND) of several different literals. For

example, if you have two variables, A and B, there are eight possible terms: A, B, A’, B’,

A’B’,A’B, AB’, and AB. For three variables we have 26 different terms: A, B, C, A’, B’, C’,

A’B’, A’B, AB’, AB, A’C’, A’C, AC’, AC, B’C’, B’C, BC’, BC, A’B’C’, AB’C’, A’BC’, ABC’,

A’B’C AB’C, A’BC, and ABC. As you can see, as the number of variables increases, the

number of terms increases dramatically. A minterm is a product containing exactly n

literals. For example, the minterms for two variables are A’B’, AB’, A’B, and AB. Likewise,

the minterms for three variables A, B, and C are A’B’C’, AB’C’, A’BC’, ABC’, A’B’C, AB’C,

A’BC, and ABC. In general, there are 2n minterms for n variables. The set of possible

minterms is very easy to generate since they correspond to the sequence of binary numbers:

Binary

Equivalent

Minterm

000 A’ B’ C’

001 A’ B’ C

010 A’ B C’

011 A’ B C

100 A B’ C’

101 A B’ C

110 A B C’

111 A B C

Table 3.9 Minterms for Three Input Variables

We can specify any Boolean function using a sum (logical OR) of minterms. Given F248

=AB+C the equivalent canonical form is ABC+A’BC+AB’C+A’B’C+ABC’. Algebraically, we

can show that these two are equivalent as follows:

ABC+A’BC+AB’C+A’B’C+ABC’ = BC (A+A’) + B’C (A+A’) + ABC’

= BC•1 +B’C•1 + ABC’

= C(B+B’) + ABC’

= C + ABC’

= C + AB

Obviously, the canonical form is not the optimal form. On the other hand, there is a big

advantage to the sum of minterms canonical form: it is very easy to generate the truth table

for a function from this canonical form. Furthermore, it is also very easy to generate the

logic equation from the truth table. To build the truth table from the canonical form, simply

convert each minterm into a binary value by substituting a “1” for unprimed variables and

a “0” for primed variables. Then place a “1” in the corresponding position (specified by the

binary minterm value) in the truth table:

1) Convert minterms to binary equivalents:

F248= CBA + CBA’ + CB’A + CB’A’ + C’BA

= 111 + 110 + 101 + 100 + 011

2) Substitute a one in the truth table for each entry above. Finally, put zeros in all the entries

that you did not fill with ones in the first step above

Table 3.10: Creating a Truth Table from Minterms, Step One

Table 3.11: Creating a Truth Table from Minterms, Step Two

Going in the other direction, generating a logic function from a truth table, is almost as

easy. First, locate all the entries in the truth table with a one. In the table above, these are

the last five entries. The number of table entries containing ones determines the number of

minterms in the canonical equation. To generate the individual minterms, substitute A, B,

or C for ones and A’, B’, or C’ for zeros in the truth table above. Then compute the sum of

these items. In the example above, F248 contains one for CBA = 111, 110, 101, 100, and 011.

Therefore, F248 = CBA + CBA’ + CB’A + CB’A’ + C’AB. The first term, CBA, comes from the

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last entry in the table above. C, B, and A all contain ones so we generate the minterm CBA

(or ABC, if you prefer). The second to last entry contains 110 for CBA, so we generate the

minterm CBA’. Likewise, 101 produce CB’A; 100 produce CB’A’ and 011 produces C’BA. Of

course, the logical OR and logical AND operations are both commutative, so we can

rearrange the terms within the minterms as we please and we can rearrange the minterms

within the sum as we see fit. This process works equally well for any number of variables.

Consider the function F53504 = ABCD + A’BCD + A’B’CD + A’B’C’D. Placing ones in the

appropriate positions in the truth table generates the following:

D C B A F = ABCD + A’BCD

+ A’B’CD +

A’B’C’D

0 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 0 0 0 1

1 0 0 1

1 0 1 0

1 0 1 1

1 1 0 0 1

1 1 0 1

1 1 1 0 1

1 1 1 1 1

Table 3.12: Creating a Truth Table with Four Variables from Minterms

The remaining elements in this truth table all contain zero.

Perhaps the easiest way to generate the canonical form of a Boolean function is to first

generate the truth table for that function and then build the canonical form from the truth

table. We’ll use this technique, for example, when converting between the two canonical

forms this chapter presents. However, it is also a simple matter to generate the sum of

minterms form algebraically. By using the distributive law and theorem 15 (A + A’ = 1)

makes this task easy. Consider F248 = AB + C. This function contains two terms, AB and C,

but they are not minterms. Minterms contain each of the possible variables in a primed or

unprimed form. We can convert the first term to a sum of minterms as follows:

AB = AB • 1 By Th4

= AB • (C + C’) By Th 15

= ABC + ABC’ By distributive law

= CBA + C’BA By associative law

Similarly, we can convert the second term in F248 to a sum of minterms as follows:

C = C • 1 By Th4

= C • (A + A’) By Th15

= CA + CA’ By distributive law

= CA•1 + CA’•1 By Th4

= CA • (B + B’) + CA’ • (B + B’) By Th15

= CAB + CAB’ + CA’B + CA’B’ By distributive law

= CBA + CBA’ + CB’A + CB’A’ By associative law

The last step (rearranging the terms) in these two conversions is optional. To obtain the

final canonical form for F248 we need only sum the results from these two conversions:

F248 = (CBA + C’BA) + (CBA + CBA’ + CB’A + CB’A’)

= CBA + CBA’ + CB’A + CB’A’ + C’BA

Another way to generate a canonical form is to use products of maxterms. A maxterm is the

sum (logical OR) of all input variables, primed or unprimed. For example, considerthe

following logic function G of three variables:

G = (A+B+C) • (A’+B+C) • (A+B’+C).

Like the sum of minterms form, there is exactly one product of maxterms for each possible

logic function. Of course, for every product of maxterms there is an equivalent sum of

minterms form. In fact, the function G, above, is equivalent to

F248 = CBA + CBA’ + CB’A + CB’A’ + C’BA = AB +C.

Generating a truth table from the product of maxterms is no more difficult than building it

from the sum of minterms. You use the duality principle to accomplish this. Remember, the

duality principle says to swap AND for OR and zeros for ones (and vice versa).

Therefore, to build the truth table, you would first swap primed and non-primed literals. In

G above, this would yield:

G= (A’ + B’ + C’) • (A + B’ + C’) • (A’ + B + C’)

The next step is to swap the logical OR and logical AND operators. This produces

G = A’B’C’ + AB’C’ + A’BC’

Finally, you need to swap all zeros and ones. This means that you store zeros into the truth

table for each of the above entries and then fill in the rest of the truth table with ones. This

will place a zero in entries zero, one, and two in the truth table. Filling the remaining

entries with ones produces F248.

You can easily convert between these two canonical forms by generating the truth table for

one form and working backwards from the truth table to produce the other form. For

example, consider the function of two variables, F7 = A + B. The sum of minterms form is

F7 = A’B + AB’ + AB. The truth table takes the form:

Working backwards to get the product of maxterms, we locate all entries that have a zero

result. This is the entry with A and B equal to zero. This gives us the first step of G=A’B’.

However, we still need to invert all the variables to obtain G=AB. By the duality principle

we need to swap the logical OR and logical AND operators obtaining G=A+B.

This is the canonical product of maxterms form. Since working with the product of maxterms

is a little easier than working with sums of minterms, this text will generally use the sum of

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minterms form. Furthermore, the sum of minterms form is more common in boolean logic

work. However, you will encounter both forms when studying logic design.

3.8.4 Simplifications of Boolean Functions:

Since there are an infinite variety of Boolean functions of n variables, but only a finite

number of unique boolean functions of those n variables, you might wonder if there is

some method that will simplify a given boolean function to produce the optimal form. Of

course, you can always use algebraic transformations to produce the optimal form, but

using heuristics does not guarantee an optimal transformation. There are, however, two

methods that will reduce a given Boolean function to its optimal form: the map method and

the prime implicants method. In this text we will only cover the mapping method, see any

text on logic design for other methods.

Table 3.13: F7 (OR) Truth Table for Two Variables

Working backwards to get the product of maxterms, we locate all entries that have a zero

result. This is the entry with A and B equal to zero. This gives us the first step of G=A’B’.

However, we still need to invert all the variables to obtain G=AB. By the duality principle

we need to swap the logical OR and logical AND operators obtaining G=A+B. This is the

canonical product of maxterms form. Since working with the product of maxterms is a little

messier than working with sums of minterms, this text will generally use the sum of

minterms form. Furthermore, the sum of minterms form is more common in boolean logic

work. However, you will encounter both forms when studying logic design.

Since for any logic function some optimal form must exist, you may wonder why we don’t

use the optimal form for the canonical form. There are two reasons. First, there may be

several optimal forms. They are not guaranteed to be unique. Second, it is easy to convert

between the canonical and truth table forms.

Using the map method to optimize boolean functions is practical only for functions of two,

three, or four variables. With care, you can use it for functions of five or six variables, but

the map method is cumbersome to use at that point. For more than six variables,

attempting map simplifications by hand would not be wise1. The first step in using the

map method is to build a two-dimensional truth table for the function.

Fig 15 Two, Three, and Four Dimensional Truth Maps

Warning: Take a careful look at these truth tables. They do not use the same forms

appearing earlier in this chapter. In particular, the progression of the values is 00, 01, 11, 10,

not 00, 01, 10, 11. This is very important! If you organize the truth tables in a binary

sequence, the mapping optimization method will not work properly. We will call this a

truth map to distinguish it from the standard truth table.

Assuming your boolean function is in canonical form (sum of minterms), insert ones for

each of the truth map entries corresponding to a minterm in the function and place zeros

every where else.

For example, consider the function of three variables

F=C’B’A + C’BA’ +C’BA + CB’A’ + CB’A + CBA’ + CBA.

Fig 16 : A Sample Truth Map

The next step is to draw rectangles around rectangular groups of ones. The rectangles you

enclose must have sides whose lengths are powers of two. For functions of three variables,

the rectangles can have sides whose lengths are one, two, and four. The set of rectangles

you draw must surround all cells containing ones in the truth map. The trick is to draw all

possible rectangles unless a rectangle would be completely enclosed within another. Note

that the rectangles may overlap if one does not enclose the other. In the truth map in Figure

2.2 there are three such rectangles (see Figure 2.3)

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Fig 17: Surrounding Rectangular Groups of Ones in a Truth Map

Each rectangle represents a term in the simplified boolean function. Therefore, the

simplified boolean function will contain only three terms. You build each term using the

process of elimination. You eliminate any variables whose primed and unprimed form both

appear within the rectangle. Consider the long skinny rectangle above that is sitting in the

row where C=1. This rectangle contains both A and B in primed and unprimed form.

Therefore, we can eliminate A and B from the term. Since the rectangle sits in the C=1

region, this rectangle represents the single literal C.

Now consider the solid square above. This rectangle includes C, C’, B, B’ and A. Therefore,

it represents the single term A. Likewise, the square with the dotted line above contains C,

C’, A, A’ and B. Therefore, it represents the single term B. The final, optimal, function is the

sum (logical OR) of the terms represented by the three squares. Therefore, F= A + B + C.

You do not have to consider squares containing zeros.

When enclosing groups of ones in the truth map, you must consider the fact that a truth

map forms a torus (i.e., a doughnut shape). The right edge of the map wraps around to the

left edge (and vice-versa). Likewise, the top edge wraps around to the bottom edge. This

introduces additional possibilities when surrounding groups of ones in a map. Consider the

boolean function F=C’B’A’ + C’BA’ + CB’A’ + CBA’. Figure 2.4 shows the truth map for this

function.

Fig 18: Truth Map for F=C’B’A’ + C’BA’ + CB’A’ + CBA’

At first glance, you would think that there are two possible rectangles here as Figure 2.5

shows. However, because the truth map is a continuous object with the right side and left

sides connected, we can form a single, square rectangle, as Figure 2.6 shows.

So what? Why do we care if we have one rectangle or two in the truth map? The answer is

because the larger the rectangles are, the more terms they will eliminate. The fewer

rectangles that we have, the fewer terms will appear in the final boolean function. For

example, the former example with two rectangles generates a function with two terms. The

first rectangle (on the left) eliminates the C variable, leaving A’B’ as its term. The second

rectangle, on the right, also eliminates the C variable, leaving the term BA’.

Therefore, this truth map would produce the equation F=A’B’ + A’B. We know this is not

optimal, see Th 13. Now consider the second truth map above. Here we have a single

rectangle so our Boolean function will only have a single term. Obviously this is more

optimal than an equation with two terms. Since this rectangle includes both C and C’ and

also B and B’, the only term left is A’. This Boolean function, therefore, reduces to F=A’.

There are only two cases that the truth map method cannot handle properly: a truth map

that contains all zeros or a truth map that contains all ones. These two cases correspond to

the boolean functions F=0 and F=1, respectively. These functions are easy to generate by

inspection of the truth map.

Fig 19: First attempt at Surrounding Rectangles Formed by Ones

Fig 20: Correct Rectangle for the Function

An important thing you must keep in mind when optimizing Boolean functions using the

mapping method is that you always want to pick the largest rectangles whose sides lengths

are a power of two. You must do this even for overlapping rectangles (unless one rectangle

encloses another). Consider the boolean function F = C'B'A' + C'BA' + CB'A' +C'AB + CBA'

+ CBA. This produces the truth map appearing in Figure 2.7.The initial temptation is to

create one of the sets of rectangles found in Figure 2.8. However, the correct mapping

appears in Figure 2.9. All three mappings will produce a Boolean function with two terms.

However, the first two will produce the expressions F= B + A'B' and F = AB + A'. The third

form produces F = B + A'. Obviously, this last form is more optimal than the other two

forms (see theorems 11 and 12).

For functions of three variables, the size of the rectangle determines the number of terms it

represents:

• A rectangle enclosing a single square represents a minterm. The associated term will

have three literals.

• A rectangle surrounding two squares containing ones represents a term containing

two literals.

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• A rectangle surrounding four squares containing ones represents a term containing a

single literal.

• A rectangle surrounding eight squares represents the function F = 1.

Truth maps you create for functions of four variables are even trickier. This is because there

are lots of places rectangles can hide from you along the edges. Figure 2.10 shows some

possible places rectangles can hide.

Fig 21: Truth Map for F = C'B'A' + C'BA' + CB'A' + C'AB + CBA' + CBA

Fig 22: Obvious Choices for Rectangles

Fig 22 Correct Set of Rectangles for F = C'B'A' + C'BA' + CB'A' + C'AB + CBA' + CBA

Figure 23: Partial Pattern List for 4x4 Truth Map

This list of patterns doesn’t even begin to cover all of them! For example, these diagrams

show none of the 1x2 rectangles. You must exercise care when working with four variable

maps to ensure you select the largest possible rectangles, especially when overlap occurs.

This is particularly important with you have a rectangle next to an edge of the truth map.

As with functions of three variables, the size of the rectangle in a four variable truth map

controls the number of terms it represents:

• A rectangle enclosing a single square represents a minterm. The associated term will

have four literals.

• A rectangle surrounding two squares containing ones represents a term containing

three literals.

• A rectangle surrounding four squares containing ones represents a term containing

two literals.

• A rectangle surrounding eight squares containing ones represents a term containing

a single literal.

• A rectangle surrounding sixteen squares represents the function F=1.

This last example demonstrates an optimization of a function containing four variables. The

function is F = D’C’B’A’ + D’C’B’A + D’C’BA + D’C’BA’ + D’CB’A + D’CBA + DCB’A +

DCBA + DC’B’A’ + DC’BA’,. Here are two possible sets of maximal rectangles for this

function; each producing three terms .Both functions are equivalent; both are as optimal as

you can get2. Either will suffice for our purposes.

First, let’s consider the term represented by the rectangle formed by the four corners. This

rectangle contains B, B’, D, and D’; so we can eliminate those terms. The remaining terms

contained within these rectangles are C’ and A’, so this rectangle represents the term C’A’.

The second rectangle, common to both maps in, is the rectangle formed by the iddle four

squares. This rectangle includes the terms A, B, B’, C, D, and D’. Eliminating B, B’, D, and

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D’ (since both primed and unprimed terms exist), we obtain CA as the term for this

rectangle. The map on the left in Figure has a third term represented by the top row.

Thisterm includes the variables A, A’, B, B’, C’ and D’. Since it contains A, A’, B, and B’, we

can eliminate these terms. This leaves the term C’D’. Therefore, the function represented by

the map on the left is F=C’A’ + CA + C’D’.

The map on the right has a third term represented by the top/middle four squares. This

rectangle subsumes the variables A, B, B’, C, C’, and D’. We can eliminate B, B’, C, and C’

since both primed and unprimed versions appear, this leaves the term AD. Therefore, the

function represented by the function on the right is F=C’A’ + CA + AD’.

Since both expressions are equivalent, contain the same number of terms, and the same

number of operators, either form is equivalent. Unless there is another reason for choosing

one over the other, you can use either form.

3.8.7 Quine-McCluskey minimization method:

Quine-McCluskey minimization method uses the same theorem to produce the solution as

the K-map method, namely X(Y+Y')=X

Minimization Technique

The expression is represented in the canonical SOP form if not already in that form. The

function is converted into numeric notation.

The numbers are converted into binary form.

The minterms are arranged in a column divided into groups.

Begin with the minimization procedure.

Each minterm of one group is compared with each minterm in the group

immediately below.

Each time a number is found in one group which is the same as a number in

the group below except for one digit, the numbers pair is ticked and a new

composite is created.

This composite number has the same number of digits as the numbers in the

pair except the digit different which is replaced by an "x".

The above procedure is repeated on the second column to generate a third column.

The next step is to identify the essential prime implicants, which can be done using a

prime implicant chart.

Where a prime implicant covers a minterm, the intersection of the

corresponding row and column is marked with a cross.

Those columns with only one cross identify the essential prime implicants.

These prime implicants must be in the final answer.

The single crosses on a column are circled and all the crosses on the same row

are also circled, indicating that these crosses are covered by the prime

implicants selected.

Once one cross on a column is circled, all the crosses on that column can be

circled since the minterm is now covered.

If any non-essential prime implicant has all its crosses circled, the prime

implicant is redundant and need not be considered further.

Next, a selection must be made from the remaining nonessential prime implicants,

by considering how the non-circled crosses can be covered best.

One generally would take those prime implicants which cover the greatest

number of crosses on their row.

If all the crosses in one row also occur on another row which includes further

crosses, then the latter is said to dominate the former and can be selected.

The dominated prime implicant can then be deleted.

Example:

Find the minimal sum of products for the Boolean expression, f= (1,2,3,7,8,9,10,11,14,15),

using Quine-McCluskey method

Firstly these minterms are represented in the binary form as shown in the table below. The

above binary representations are grouped into a number of sections in terms of the number

of 1's as shown in the table below

Table 3.14 Binary representation of minterms

Minterm U V W X

1 0 0 0 1

2 0 0 1 0

3 0 0 1 1

7 0 1 1 1

8 1 0 0 0

9 1 0 0 1

10 1 0 1 0

11 1 0 1 1

14 1 1 1 0

15 1 1 1 1

No of

1’s

Minterm U V W X

1 1 0 0 0 1

1 2 0 0 1 0

1 8 1 0 0 0

2 3 0 0 1 1

2 9 1 0 0 1

2 10 1 0 1 0

3 7 0 1 1 1

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Table 3.15: Group of minterms for different number of 1's

Any two numbers in these groups which differ from each other by only one variable can be

chosen and combined, to get 2-cell combination, as shown in the table below

Combinations U V W X

(1,3) 0 0 - 1

(1,9) - 0 0 1

(2,3) 0 0 1 -

(2,10) - 0 1 0

(8,9) 1 0 0 -

(8,10) 1 0 - 0

(3,7) 0 - 1 1

(3,11) - 0 1 1

(9,11) 1 0 - 1

(10,11) 1 0 1 -

(10,14) 1 - 1 0

(7,15) - 1 1 1

(11,15) 1 - 1 1

(14,15) 1 1 1 -

Table 3.16 2-Cell combinations

From the 2-cell combinations, one variable and dash in the same position can be combined

to form 4-cell combinations as shown in the figure below:

Table 3.17 4-Cell combinations

The cells (1,3) and (9,11) form the same 4-cell combination as the cells (1,9) and (3,11). The

order in which the cells are placed in a combination does not have any effect. Thus the

(1,3,9,11) combination could be written as (1,9,3,11).

From above 4-cell combination table, the prime implicants table can be plotted as shown in

table below

3 11 1 0 1 1

3 14 1 1 1 0

4 15 1 1 1 1

Combinations U V W X

(1,3,9,11) - 0 - 1

(2,3,10,11) - 0 1 -

(8,9,10,11) 1 0 - -

(3,7,11,15) - - 1 1

(10,11,14,15) 1 - 1 -

Prime

Implicants 1 2 3 7 8 9 10 11 14 15

(1,3,9,11) X - X - - X - X - -

(2,3,10,11) - X X - - - X X - -

(8,9,10,11) - - - - X X X X - -

(3,7,11,15) - - - - - - X X X X

- X X - X X - - - X -

Table 3.18 Prime Implicants Table

The columns having only one cross mark corresponds to essential prime implicants. A

yellow cross is used against every essential prime implicant. The prime implicants sum

gives the function in its minimal SOP form

Y = V'X + V'W + UV' + WX + UW

Example: Minimize the following expression using Quine McClusky method.

f (A, B, C, D) = ∑ m(2, 9, 10, 11, 13, 15)

Solution:

Step 1: First list out all minterms

Minterms Binary Equivalent

A B C D

m2 0 0 1 0

m9 1 0 0 1

m10 1 0 1 0

m11 1 0 1 1

m13 1 1 0 1

m15 1 1 1 1

Step 2: Arrange minterms depending upon number of 1’s from above table.

Grouping Minterms Binary Equivalent

A B C D

1 m2 0 0 1 0

2 m9

m10

1

1

0

0

0

1

1

0

3 m11

m13

1

1

0

1

1

0

1

1

4 m15 1 1 1 1

Step 3: Prepare first reduction table.


A B C D

1 m2, m10 X 0 1 0 Pa

2 m9, m11

m9, m13

1

1

0

X

X

0

1

1

√

√

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m10, m11 1 0 1 X Pb

3 m11, m15

m13, m15

1

1

X

1

1

X

1

1

√

√

Step 4: Now prepare another reduction table


A B C D

2 m9, m11, m13,

m15

m9, m13, m11,

m15

1

1

X

X

X

X

1

1

Pc

Pc

Step 5: Now prepare prime implicants table

Prime

Implicants

Minterms

m2 m9 m10 m11 m13

m15

Pa = X 0 1 0 X X

Pb = 1 0 1 X X

Pc = 1 X X 1 X X X X

Pc is an essential implicant because it covers minterms 9, 11, 13, 15. Remaining terms are Pa

and Pb covers minterms 2, 10, 11. But minterm 11 is already covered by Pc therefore

consider only minterms 2 and 10 by selecting Pa

f (A, B, C, D) = [X 0 1 0, 1 X X 1] = B’CD’ + AD

3.9. Short Answer Questions:

1. What are universal gates? Ans: A universal gate is a gate which can implement any Boolean function without need to

use any other gate type. The NAND and NOR gates are universal gates. In practice, this is

advantageous since NAND and NOR gates are economical and easier to fabricate and are

the basic gates used in all IC digital logic families. In fact, an AND gate is typically

implemented as a NAND gate followed by an inverter not the other way around!! Likewise,

an OR gate is typically implemented as a NOR gate followed by an inverter not the other

way around.

2. What is Standard POS form?

Ans: If each term in POS form contains all the literals then the POS form is known as

standard or canonical POS form.

3. What is Standard SOP form?

Ans: If each term in SOP form contains all the literals then the SOP form is known as

standard or canonical SOP form.

4. What is Two-level realization?

Ans: Realization of logic functions in which signals pass through two gates, such as AND-

OR, OR-AND,NAND-NAND,NOR-NOR realization.

5.What is combinational logic?

Ans: The logic in which the output at any instant of time is dependent only on the inputs

presents at that time

6.What is don’t care condition?

Ans. A minterm /maxterm in a logic function which may or may not be included.

3.8.2 Subjective Type Questions:

1) Simplify the following :A+ B+ C+ D Ans. A+B+C+D

2) Simplify the following: A [B+C ( )]. Ans. AB

3) Prove the following: BC+A C+AB +ABC=AB+BC+AC 4) Prove the following:

A B = A = B 5) State and prove DeMorgan’s theorem. 6) Simplify the following:

AB+ABC+A(B+A ) Ans.0 7) Prove the following using DeMorgan’s theorem AB+CD = AB . CD 8) State and prove the DeMorgan’s theorem 9) Simplify following expression using K-map and realize using AND-OR realization f(A,B,C,D)= ∑m(1,3,7,8,10,12,13,15)

10) Obtain K-map for f=A B+A C + C +A D + ABC + A B C a) Express it in standard SOP form. b) Minimize it and realize the minimized expression using NAND gates only

3.9 References:

1.R. P. Jain, “Modern Digital Electronics”, Tata McGraw Hill.

2. M. Morris Mano, “Digital Logic and computer Design”, PHI.

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3. Anil K. Maini, “Digital Electronics Principles and Integrated Circuits”, Wiley

India

Self-evaluation

Name of

Student

Class SE

Roll No.

Subject Logic Design

Module No. 03

S.No Tick

Your choice

1. Do you understand standard SOP & POS forms of

boolean expression & their reduction ?

o Yes

o No

2. Do you understand the different types reduction

methods ?

o Yes

o No

3. Do you understand basic digital circuits ? o Yes

o No

4. Do you understand boolean algebra and theorems?

o Yes

o No

5. Do you understand module ?

Yes, Completely.

Partialy.

No, Not at all.

Boolean Algebra and Logic gates · easier to fabricate and are the basic gates used in all IC...

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