De Morgan's Laws

4 Key excerpts on "De Morgan's Laws"

• 

  eBook - ePub

  Electronic Logic Circuits

  • J. Gibson(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  must be identical. This demonstrates that de Morgan’s theorem is true in this particular case.

  The relationships of Boolean algebra may be used to manipulate Boolean expressions (that is mathematical descriptions of logic circuits) into alternative forms. The usual reason for performing such manipulations is to change one Boolean expression for a circuit into another which is more easily or more economically constructed from the available components.

  Example 2.3 Suppose the output, R, of some circuit is given as Can this be manipulated into a more simple form?

  Solution

  Change the expression using the Distributive laws to

  Since B + = 1 from the Complement relations, then it can be written as

  which can be further treated in the same way giving This final form is much simpler than the original, but some steps in the reduction were not obvious unless the answer was already known.

  One fault of Boolean algebra is that the steps in the manipulation are not always obvious. Chapter 3 introduces some methods which may be used to reduce complicated Boolean expressions in a reliable and consistent manner.

  2.5 De Morgan’s theorem De Morgan’s theorem (or law) is very important; it is probably one of the most frequently used relationships in Boolean algebra. The two forms are and both forms are valid with any number of variables.

  The principle use of de Morgan’s theorem is to convert an OR type of expression (i.e. either OR or NOR) into an AND form (i.e. either AND or NAND) and vice versa when a particular type of logic gate is to be used for circuit construction. For example, suppose that X = A + B + C and that a circuit is required to produce X using only NAND gates. Because double inversion is the same as no inversion, then X = ; applying de Morgan’s theorem to

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  eBook - ePub

  Digital Electronic Circuits

  Principles and Practices

  • Shuqin Lou, Chunling Yang(Authors)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  In Boolean algebra, there are certain well-developed laws, rules, and theorems that must be followed in order to properly apply Boolean algebra. This section only introduces the most important Boolean algebra laws, rules, and theorems for analyzing and designing digital circuits.

  The objectives of this section are to

  • – Apply commutative laws, associative laws, and distributive laws
  • – Apply basic rules of Boolean algebra
  • – Apply DeMorgan’s theorems

  3.3.1 Boolean addition and Boolean multiplication

  1. Boolean addition

  Boolean addition is equivalent to the OR operation and the basic rules are illustrated with their relation to the OR gate as shown in Figure 3.3.1 .

  Figure 3.3.1: Illustration of the relation between Boolean addition and OR gate.

  In Boolean algebra, a sum term is a sum of literals. In logic circuit, a sum term could be produced by an OR operation without AND operation involved. For example, A +C ,

  A +

  B ˉ

  and

  A ˉ

  +

  B ˉ

  +

  C ˉ

  are sum terms.

  2. Boolean multiplication

  Boolean multiplication is equivalent to AND operation and the basic rules are illustrated with their relation to AND gate as shown in Figure 3.3.2 .

  Figure 3.3.2: Illustration of the relation between Boolean multiplication and AND gate.

  In Boolean algebra, a product term is a product of literals. In logic circuit, a product term could be produced by an AND operation without OR operation involved. For example, AC ,

  A

  B ˉ

  ,

  A ˉ

  B ˉ

  C ˉ

  , and

  A ˉ

  B ˉ

  C D

  are product terms.

  3.3.2 Laws of Boolean algebra

  Similar to ordinary algebra, Boolean algebra has three basic laws including commutative laws, associative laws, and distributive laws. Each of the laws is illustrated with two or three variables, but the number of variables can be extended.

  1. Commutative laws

  The commutative laws for the multiplication and the addition of two variables are expressed as below:

  A B = B A

  A + B = B + A

  2. Associative laws

  The associative laws for the multiplication and the addition of three variables are written as follows:

  A B

  C = A

  B C

  A + B

  + C = A +

  B + C

  3. Distributive laws

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  eBook - ePub

  Computer Fundamentals - 8th Edition

  Concepts, Systems & Applications

  • Pradeep K.Sinha(Author)
  • 2004(Publication Date)
  • BPB Publications

    (Publisher)

  HAPTER 6

  Boolean Algebra and Logic Circuits

  Boolean algebra deals with binary number system. It is very useful in designing logic circuits used in processors of computer systems. In this chapter, you will learn about this algebra and elementary logic gates used to build up logic circuits of different types for performing necessary arithmetic operations. These logic gates are the building blocks of all logic circuits in a computer. You will also learn how to use Boolean algebra for designing simple logic circuits used frequently by arithmetic logic unit of almost all computers.

  BOOLEAN ALGEBRA

  In mid 1800, George Boole (1815-1864), an English mathematician, developed algebra for simplifying the representation and manipulation of propositional logic. It is known as Boolean algebra after its developer's name. Later, in the year 1938, Claude E. Shannon proposed the use of Boolean algebra in the design of relay switching circuits. The basic techniques described by Shannon were adopted almost universally for the design and analysis of switching circuits. Owing to analogous relationship between the action of relays and modern electronic circuits, designers of modern computers still use the same techniques.

  Boolean algebra provides an economical and straightforward approach to the design of relay and other types of switching circuits. Just as basic theorems of algebra help in simplifying an ordinary algebraic expression, Boolean algebra helps in simplifying an expression describing a given switching circuit. Today, designers use Boolean algebra extensively in designing electronic circuitry of computers.

  Fundamental Concepts of Boolean Algebra

  Fundamental concepts of Boolean algebra are described below.

  Use of Binary Digits

  In a normal algebraic expression, a variable can take any numerical value. For example, in the expression 3A + 7B = C, each of the variables A, B, and C may have any value from the entire field of real numbers. Since, Boolean algebra deals with binary number system, the variables used in Boolean equations may have only two possible values (0 or 1). For example, in the Boolean equation A + B = C, each of the variables A, B, and C may have only the values 0 or 1.

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  eBook - ePub

  Digital Systems Design, Volume I

  Numbering Systems and Logical Operations

  • Larry Massengale(Author)
  • 2019(Publication Date)
  • Momentum Press

    (Publisher)

  CHAPTER 3

  BOOLEAN ALGEBRA

  3.1.INTRODUCTION

  What is Boolean Algebra? According to Merriam–Webster,

  Boolean Algebra is an algebraic system that consists of a set closed under two binary operations and that can be described by any of various systems of postulates all of which can be deduced from the postulates that each operation is commutative, that each operation is distributive over the other, that an identity element exists for each operation, and that for every element in the set there exists another element which when combined with the first under either one of the operations yields the identity element of the other operation.

  Modern Boolean Algebra is known to be derived from the work of philosopher George Boole. This work comes primarily from his two books, The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854). His work is also referred to as Analysis and Laws, which consist of mathematical symbols and variables.

  Boolean Algebra has been essential in the development of digital electronics. As stated in Chapters 1 and 2 , in the binary (base 2) numbering system, the two logic levels used to represent lows and highs are Logic 0 and Logic 1, respectively. However, Boolean Algebra symbols such as 0 or 1 are not numbers but values of switching variables. Symbol 0 could represent a high voltage, while symbol 1 could represent a low voltage. Knowing this, logic circuits allow for the use of Boolean Algebra as a means for analyzing and designing digital electronics and systems. This chapter will describe how Boolean Algebra is used for these operations.

  3.2.BOOLEAN SYMBOLS

  The mathematical symbols used in Boolean Algebra are shown in Table 3.1 . These mathematical symbols represent the logical functions accomplished by digital circuits and systems.

  Table 3.1. Symbology and meaning

  3.3.BASIC LOGIC OPERATIONS

  3.3.1.AND LOGIC

  The AND gate shown in Figure 3.1 produces an output of A ∙ B. This equation represents a logical product of the circuit shown. The product is developed by ANDing two or more inputs together. The product for this gate can also be written as AB. So the output Boolean expression would be x = A ∙ B. Using the truth table in Figure 3.2

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