# Boolean algebra and its applications by j eldon whitesitt pdf

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In mathematics and mathematical logic , Boolean algebra is the subarea of algebra in which the values of the variables are the truth values true and false , usually denoted 1 and 0 respectively. Boolean algebra has been fundamental in the development of computer science and is yet the basis of the abstract description of digital circuits. It is also used in digital logic, computer programming , set theory , social science [ 3 ] , and statistics. Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields.

In fact, M. Stone proved in that every Boolean algebra is isomorphic to a field of sets. In the s, while studying switching circuits, Claude Shannon observed that one could also apply the rules of Boole's algebra in this setting, and he introduced switching algebra as a way to analyze and design circuits by algebraic means in terms of logic gates.

Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra. In circuit engineering settings today, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably. Modern electronic design automation tools for VLSI circuits often rely on an efficient representation of Boolean functions known as reduced ordered binary decision diagrams BDD for logic synthesis and formal verification.

Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way.

Although the development of mathematical logic did not follow Boole's program, the connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other logics. The closely related model of computation known as a Boolean circuit relates time complexity of an algorithm to circuit complexity. Whereas, in elementary algebra, the variables represent mainly numbers, in Boolean algebra, the variables represent the truth values false and true , which are identified with the bits or binary digits and denoted by 0 and 1.

They should not be confused with the integers 0 and 1, but may also identified with the elements of the field with two elements. A sequence of bits is a commonly used such function. Another common example is the subsets of a set E : to a subset F of E is associated the indicator function that takes the value 1 on E and 0 outside E.

Like in usual algebra, most of the theory may be developed without considering explicit values for the variables. If the truth values 0 and 1 are interpreted as integers, these operation may be expressed with the ordinary operations of the arithmetic:. One may consider that only the negation and one of the two other operations are basic, because of the following identities that allow to define the conjunction in terms of the negation and the disjunction, and vice versa:.

We have so far seen three Boolean operations. We referred to these as basic, meaning that they can be taken as a basis for other Boolean operations that can be built up from them by composition, the manner in which operations are combined or compounded.

Here are some examples of operations composed from the basic operations. These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs. But if x is false then we ignore the value of y ; however we must return some truth value and there are only two choices, so we choose the value that entails less, namely true. Relevance logic addresses this by viewing an implication with a false premise as something other than either true or false.

It excludes the possibility of both x and y. In particular the following laws are common to both kinds of algebra: [ 14 ]. Boolean algebra however obeys some additional laws, in particular the following: [ 14 ].

All of the laws treated so far have been for conjunction and disjunction. These operations have the property that changing either argument either leaves the output unchanged or the output changes in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0.

Operations with this property are said to be monotone. Thus the axioms so far have all been for monotonic Boolean logic. All properties of negation including the laws below follow from the above two laws alone.

In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, whence in both algebras it satisfies the double negation law also called involution law. Boolean algebra satisfies De Morgan's laws ,. At this point we can now claim to have defined Boolean algebra, in the sense that the laws we have listed up to now entail the rest of the subject. The laws Complementation 1 and 2, together with the monotone laws, suffice for this purpose and can therefore be taken as one possible complete set of laws or axiomatization of Boolean algebra.

Every law of Boolean algebra follows logically from these axioms. Furthermore Boolean algebras can then be defined as the models of these axioms as treated in the section thereon. To clarify, writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them. Had we stopped listing laws too soon, there would have been Boolean laws that did not follow from those on our list, and moreover there would have been models of the listed laws that were not Boolean algebras.

This axiomatization is by no means the only one, or even necessarily the most natural given that we did not pay attention to whether some of the axioms followed from others but simply chose to stop when we noticed we had enough laws, treated further in the section on axiomatizations. Or the intermediate notion of axiom can be sidestepped altogether by defining a Boolean law directly as any tautology , understood as an equation that holds for all values of its variables over 0 and 1.

All these definitions of Boolean algebra can be shown to be equivalent. There is nothing magical about the choice of symbols for the values of Boolean algebra. But suppose we rename 0 and 1 to 1 and 0 respectively. Then it would still be Boolean algebra, and moreover operating on the same values. So there are still some cosmetic differences to show that we've been fiddling with the notation, despite the fact that we're still using 0s and 1s.

But if in addition to interchanging the names of the values we also interchange the names of the two binary operations, now there is no trace of what we have done.

The end product is completely indistinguishable from what we started with. When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, we call the members of each pair dual to each other. The Duality Principle , also called De Morgan duality , asserts that Boolean algebra is unchanged when all dual pairs are interchanged.

One change we did not need to make as part of this interchange was to complement. We say that complement is a self-dual operation.

The identity or do-nothing operation x copy the input to the output is also self-dual. It can be shown that self-dual operations must take an odd number of arguments; thus there can be no self-dual binary operation. The principle of duality can be explained from a group theory perspective by fact that there are exactly four functions that are one-to-one mappings automorphisms of the set of Boolean polynomials back to itself: the identity function, the complement function, the dual function and the contradual function complemented dual.

These four functions form a group under function composition, isomorphic to the Klein four-group, acting on the set of Boolean polynomials. A Venn diagram [ 16 ] is a representation of a Boolean operation using shaded overlapping regions. There is one region for each variable, all circular in the examples here. The interior and exterior of region x corresponds respectively to the values 1 true and 0 false for variable x.

The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 some authors use the opposite convention. While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a white box and a dark box, neither one containing a circle.

However we could put a circle for x in those boxes, in which case each would denote a function of one argument, x , which returns the same value independently of x , called a constant function. As far as their outputs are concerned, constants and constant functions are indistinguishable; the difference is that a constant takes no arguments, called a zeroary or nullary operation, while a constant function takes one argument, which it ignores, and is a unary operation.

Venn diagrams are helpful in visualizing laws. The result is the same as if we shaded that region which is both outside the x circle and outside the y circle, i. Digital logic is the application of the Boolean algebra of 0 and 1 to electronic hardware consisting of logic gates connected to form a circuit diagram. Each gate implements a Boolean operation, and is depicted schematically by a shape indicating the operation. The shapes associated with the gates for conjunction AND-gates , disjunction OR-gates , and complement inverters are as follows.

The lines on the left of each gate represent input wires or ports. The value of the input is represented by a voltage on the lead. For so-called "active-high" logic 0 is represented by a voltage close to zero or "ground" while 1 is represented by a voltage close to the supply voltage; active-low reverses this. The line on the right of each gate represents the output port, which normally follows the same voltage conventions as the input ports.

Complement is implemented with an inverter gate. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port. There being eight ways of labeling the three ports of an AND-gate or OR-gate with inverters, this convention gives a wide range of possible Boolean operations realized as such gates so decorated.

Not all combinations are distinct however: any labeling of AND-gate ports with inverters realizes the same Boolean operation as the opposite labeling of OR-gate ports a given port of the AND-gate is labeled with an inverter if and only if the corresponding port of the OR-gate is not so labeled.

This follows from De Morgan's laws. If we complement all ports on every gate, and interchange AND-gates and OR-gates, as in Figure 4 below, we end up with the same operations as we started with, illustrating both De Morgan's laws and the Duality Principle.

Note that we did not need to change the triangle part of the inverter, illustrating self-duality for complement. Because of the pairwise identification of gates via the Duality Principle, even though 16 schematic symbols can be manufactured from the two basic binary gates AND and OR by furnishing their ports with inverters circles , they only represent eight Boolean operations, namely those operations with an odd number of ones in their truth table.

Altogether there are 16 binary Boolean operations, the other eight being those with an even number of ones in their truth table, namely the following. The term "algebra" denotes both a subject, namely the subject of algebra , and an object, namely an algebraic structure. Whereas the foregoing has addressed the subject of Boolean algebra, this section deals with mathematical objects called Boolean algebras, defined in full generality as any model of the Boolean laws.

We begin with a special case of the notion definable without reference to the laws, namely concrete Boolean algebras, and then give the formal definition of the general notion.

A concrete Boolean algebra or field of sets is any nonempty set of subsets of a given set X closed under the set operations of union , intersection , and complement relative to X. As an aside, historically X itself was required to be nonempty as well to exclude the degenerate or one-element Boolean algebra, which is the one exception to the rule that all Boolean algebras satisfy the same equations since the degenerate algebra satisfies every equation.

Hence modern authors allow the degenerate Boolean algebra and let X be empty. Example 1. The power set 2 X of X , consisting of all subsets of X. Here X may be any set: empty, finite, infinite, or even uncountable.

Example 2. The empty set and X. This two-element algebra shows that a concrete Boolean algebra can be finite even when it consists of subsets of an infinite set.

## Algebra / Prealgebra

Embed Size px x x x x Boolean Algebra and Its Applications by J. Eldon WhitesittReview by: Thomas H. Mott, Jr. The Journal of Symbolic Logic, Vol.

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## Boolean algebra

In mathematics and mathematical logic , Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false , usually denoted 1 and 0, respectively. It is thus a formalism for describing logical operations , in the same way that elementary algebra describes numerical operations. It is also used in set theory and statistics.

Eldon Whitesitt. This Dover edition, first published in , is an unabridged republication of the Dover edition of the work originally published in by the Addison-Wesley Publishing Company, Reading, Massachusetts. George Boole — introduced in his book The Laws of Thought the first systematic treatment of logic and developed for this purpose the algebraic system now known by his name, Boolean algebra. Few mathematical works of the past years have had a greater impact upon mathematics and philosophy than this famous book. The significance of the work has been well expressed by Augustus De Morgan :.

*Written in English. The history of English literature, with an outline of the origin and growth of the English language. In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 d of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and.*

### boolean algebra and its applicationsby j. eldon whitesitt

In mathematics and mathematical logic , Boolean algebra is the subarea of algebra in which the values of the variables are the truth values true and false , usually denoted 1 and 0 respectively. Boolean algebra has been fundamental in the development of computer science and is yet the basis of the abstract description of digital circuits. It is also used in digital logic, computer programming , set theory , social science [ 3 ] , and statistics. Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields. In fact, M.

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below! Eldon Whitesitt. Library of Congress Catalog Card No. Few mathe- matical works of the past years have had a greater impact upon mathematics and philosophy than this famous book.

Unit 1 - Digital logic fundamentals: Number systems Boolean algebra gates simplification. Course: i Here's the decimal number system as an example:. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. By Dave We count and do math using decimal numbers. For example, the number A potential problem is that a string of ones and zeros could easily be a decimal numbe. Boolean Algebra Practice Problems: If Boolean function has only one term then implement by observation. Like real-number algebra, Boolean algebra is subject to certain rules which may be applied in the task of simplifying reducing expressions.

Download Citation | Whitesitt J. Eldon. Boolean Eldon. Boolean algebra and its applications. Addison-Wesley Request Full-text Paper PDF.

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Embed Size px x x x x Boolean Algebra and Its Applications by J. Eldon WhitesittReview by: Thomas H. Mott, Jr. The Journal of Symbolic Logic, Vol.

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