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Solution of equations in Boolean algebra | SpringerLink

Boolean Algebra is used to analyze and simplify the digital logic circuits. It uses only the binary numbers i. It is also called as Binary Algebra or logical Algebra. Boolean algebra was invented by George Boole in Complement of a variable is represented by an overbar. Quesgions, complement of variable B is represented as. Logical ANDing of the two or more variable is represented by writing a dot between them such as A.

Sometime the dot may be omitted like ABC. Any binary operation which satisfies the following expression boolean algebra questions and solutions referred to as commutative operation. Commutative law states that changing the sequence of the variables does not questoins any effect on the output of a logic circuit.

This law states that the order in which the logic operations are performed is irrelevant as their effect is the. This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable. Boolean Algebra Advertisements. Previous Page. Next Boolean algebra questions and solutions. Previous Page Print Page. Dashboard Logout.

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Generates false only if both the inputs are true. Generates true only if both the inputs are false. The truth-table of NAND gate is:. The truth-table of NOR gate is:. The X-OR gate symbol is given below:.

The truth table of X-OR gate is given below:. Its output is "true" if the inputs are the same and "false" if the inputs are different.

The X-NOR gate symbol is given below:. The truth table of X-NOR gate is given below:. Boolean algebra is a study of mathematical operations performed on certain variables called binary variables that can have only two values: true represented by 1 or false represented by 0. It is denoted by. OR Gate: OR gate generates true if at least any one of the input is true, otherwise it generates false output.

NOT Gate: It is also known as inverter. It inverts the input state from true to false and vice versa. Graphically it is represented by:. There are 2 De Morgan's laws or theorems:. Theorem 1: The complement of a sum of variables is equal to the product of the complement of each variables. B' thus proved. Theorem 2: The complement of a product of variables is equal to the sum of the complement of each variables. Here, A. Duality principle state can be obtained by replacing AND.

For example, duality of the expression A. Here, C. Here, 1. Associative law states that when ORing or ANDing more than two variables, the result is the same regardless of the grouping of the variables. This was formally achieved by letting:.

In Boole also published his first paper on invariants, a paper that would strongly influence Eisenstein, Cayley, and Sylvester to develop the subject. Arthur Cayley � , the future Sadlerian Professor in Cambridge and one of the most prolific mathematicians in history, wrote his first letter to Boole in , complimenting him on his excellent work on invariants. He became a close personal friend, one who would go to Lincoln to visit and stay with Boole in the years before Boole moved to Cork, Ireland.

In Boole started a correspondence with Augustus De Morgan � that initiated another lifetime friendship. In the schoolmaster Boole finished a lengthy paper on differential equations, combining an exponential substitution and variation of parameters with the separation of symbols method. The next year Boole read a paper at the annual meeting of the British Association for the Advancement of Science at Cambridge in June This led to new contacts and friends, in particular William Thomson � , the future Lord Kelvin.

Not long after starting to publish papers, Boole was eager to find a way to become affiliated with an institution of higher learning. He considered attending Cambridge University to obtain a degree, but was counselled that fulfilling the various requirements would likely seriously interfere with his research program, not to mention the problems of obtaining financing.

Finally, in , he obtained a professorship in a new university opening in Cork, Ireland. In the years he was a professor in Cork � he would occasionally inquire about the possibility of a position back in England. In addition Boole published 24 more papers on traditional mathematics during this period, while only one paper was written on logic, that being in He was awarded an honorary LL.

During the last 10 years of his career, from to , Boole published 17 papers on mathematics and two mathematics books, one on differential equations and one on difference equations. Both books were highly regarded, and used for instruction at Cambridge.

Also during this time significant honors came in:. Unfortunately his keen sense of duty led to his walking through a rainstorm in late , and then lecturing in wet clothes.

Not long afterwards, on December 8, in Ballintemple, County Cork, Ireland, he died of pneumonia, at the age of Another paper on mathematics and a revised book on differential equations, giving considerable attention to singular solutions, were published post mortem. The 19th century opened in England with mathematics in the doldrums. One of the obstacles to overcome in updating English mathematics was the fact that the great developments of algebra and analysis had been built on dubious foundations, and there were English mathematicians who were quite vocal about these shortcomings.

In ordinary algebra, it was the use of negative numbers and imaginary numbers that caused concern. The first major attempt among the English to clear up the foundation problems of algebra was the Treatise on Algebra , , by George Peacock � He divided the subject into two parts, the first part being arithmetical algebra , the algebra of the positive numbers which did not permit operations like subtraction in cases where the answer would not be a positive number.

The second part was symbolical algebra , which was governed not by a specific interpretation, as was the case for arithmetical algebra, but solely by laws. In symbolical algebra there were no restrictions on using subtraction, etc. The terminology of algebra was somewhat different in the 19th century from what is used today.

In this article a prefix will sometimes be added, as in number symbol or class symbol , to emphasize the intended interpretation of a symbol. Peacock believed that in order for symbolical algebra to be a useful subject its laws had to be closely related to those of arithmetical algebra.

In this connection he introduced his principle of the permanence of equivalent forms , a principle connecting results in arithmetical algebra to those in symbolical algebra. This principle has two parts:. This application of algebra captured the interest of Gregory who published a number of papers on the method of the separation of symbols , that is, the separation into operators and objects, in the CMJ.

Unfortunately these laws fell far short of what is required to justify even some of the most elementary results in algebra, like those involving subtraction.

The footnote:. In early he was stimulated to launch his investigations into logic by a trivial but very public dispute between De Morgan and the Scottish philosopher Sir William Hamilton � �not to be confused with his contemporary the Irish mathematician Sir William Rowan Hamilton � This dispute revolved around who deserved credit for the idea of quantifying the predicate e.

Within a few months Boole had written his 82 page monograph, Mathematical Analysis of Logic , giving an algebraic approach to Aristotelian logic, then looking briefly at the general theory. It essentially referred to the modern version of the algebra of logic, introduced in by William Stanley Jevons � , a version that Boole had rejected in their correspondence�see Section 5.

In MAL , and more so in LT , Boole was interested in the insights that his algebra of logic gave to the inner workings of the mind. This pursuit has met with little favor, and will not be discussed in this article. MAL , p. With his extended version of Aristotelian logic in mind where contraries enjoy equal billing , he gave MAL , p.

Regarding syllogisms, Boole did not care for the Aristotelian classification into Figures and Moods as it seemed rather arbitrary and not well-suited to the algebraic setting. In particular he did not like the requirement that the predicate of the conclusion had to be the major term in the premises. It is somewhat curious that when it came to analyzing categorical syllogisms, it was only in the conclusion that he permitted his generalized categorical propositions to appear.

Among the vast possibilities for hypothetical syllogisms, the ones that he discussed were standard, with one new example added. One can simply replace the elective symbols by their corresponding class symbols and have the interpretation used in LT in In modern terminology, this is the composition of the two operators.

This was the first mention of addition in MAL. Thus addition is a partial operation on elective operators. Likewise one finds that subtraction is defined by:. Boole added MAL , p. After stating the above distributive and commutative laws, Boole believed he was entitled to fully employ the ordinary algebra of his time, saying MAL , p.

Also any equational argument. Similar results hold for polynomials in any number of variables MAL , pp. But it is not the case that every pair of classes is disjoint.

This is the first appearance of subtraction in MAL. Then in the next several pages he adds supplementary expressions; of these the main ones will be called the secondary expressions. This was the first appearance of 0 in MAL. It was not introduced as the symbol for the empty class�indeed the empty class does not appear in MAL.

In LT , the empty class was introduced, and denoted by 0. The simple algebra and considerable detail in this part of MAL can be appealing to the new reader, but there are complications that need to be dealt with.

Syllogistic reasoning is just an exercise in elimination , namely the middle term is eliminated from the premises to give the conclusion. One finds, using the improved reduction and elimination theorems of LT , that the best possible result of elimination is. The primary equational expressions were not sufficient to derive all of the desired syllogisms. Boole introduced the alternative equational expressions see MAL , p.

Boole did not offer an algebraic way to completely determine which premises could be completed to valid syllogisms. Toward the end of the chapter on categorical syllogisms there is a long footnote MAL , pp. The footnote loses much of its force because the results it presents depend heavily on the weak elimination theorem being best possible, which is not the case.

In the Postscript he says that using only the secondary translations is altogether superior to what was presented in the main text.

Boole would use only the secondary translations of MAL in LT , but in LT the reader will no longer find a leisurely and detailed treatment of Aristotelian logic. Boole analyzed the seven hypothetical syllogisms that were standard in Aristotelian logic, from the Constructive and Destructive Conditionals to the Complex Destructive Dilemma.

This was based on adopting the standard reduction of hypothetical propositions to propositions about classes by letting the hypothetical universe , also denoted by 1, be the collection of all cases and conjunctures of Maths Vector Questions And Solutions Technology circumstances which was usually abbreviated to just the word cases.

However he makes the remark MAL , p. Boole says the universe of a categorical proposition has two cases, true and false. To find an equational expression for a hypothetical proposition Boole resorts to a near relative of truth tables MAL , p. His algebraic method of analyzing hypothetical syllogisms was to transform each of the hypothetical premises into an elective equation, and then apply his algebra of logic which was developed for categorical propositions.

Boole only considered rather simple hypothetical propositions on the grounds these were the only ones encountered in common usage see LT , p. His algebraic approach to propositional logic is easily extended to all propositional formulas as follows. Otherwise his proof is correct. This result generalizes to functions of several variables. It will not be stated as such in LT , but will be absorbed in the much more general if somewhat opaquely stated result that will be called the Rule of 0 and 1.

Constituent equations are totally interpretable. Boole shows MAL , p. Furthermore this leads MAL , p. The following table gives the constituents and modulii of their expansions:. It was natural for Boole to want to solve equations in his algebra of logic since this had been a main goal of ordinary algebra, and had led to many difficult questions e. Fortunately for Boole, the situation in his algebra of logic was much simpler�he could always solve an equation, and finding the solution was important to applications of his system, to derive conclusions in logic.

An equation was solved in part by using formal expansion after performing formal division, and then decoding the fractional coefficients. This Solution Theorem was the result of which he was the most proud�it described how to solve an elective equation for one of its symbols in terms of the others, often introducing constraint equations on the independent variables, and it is this that Boole claimed in the Introduction chapter of MAL , p.

In LT Boole would continue to regard this tool as the highlight of his work. It is superseded by the sum of squares reduction LT , p. The Elimination Theorem that he simply borrowed from algebra turned out to be weaker than what his algebra offered, and his method of reducing equations to a single equation was clumsier than the main one used in LT , but the Expansion Theorem and Solution Theorem were the same.

Much of LT would be devoted to clarifying and correcting what was said in MAL , and providing more substantial applications, the main one being his considerable work in probability theory. The second half of this page book presented probability theory as an excellent topic to illustrate the power of his algebra of logic.

Boole discussed the theoretical possibility of using probability theory enhanced by his algebra of logic to uncover fundamental laws governing society by analyzing large quantities of social data by large numbers of human computers.

Addition was introduced as aggregation when the classes were disjoint. The associative laws for addition and multiplication were conspicuously absent. A possible reason for this omission was that he worked with the standard algebra of polynomials , where the parentheses involved in the associative laws are absent, instead of the terms which are fundamental to modern logic.

Boole seems to justify his choice of laws on the basis that they are valid where defined. Working with partial algebras has its subtleties. One might expect that Boole was building toward an axiomatic foundation for his algebra of logic, just as in MAL , evidently having realized that the three laws in MAL were not enough.

Indeed he did discuss the rules of inference, that adding or subtracting equals from equals gives equals, and multiplying equals by equals gives equals. But then the development of an axiomatic approach came to an abrupt halt. There was no discussion as to whether the stated axioms which he called laws and rules which he called axioms were sufficient to construct his algebra of logic.

They were not. Instead he simply and briefly, with remarkably little fanfare, presented a radically new foundation for his algebra of logic LT pp. He said that since the only idempotent numbers were 0 and 1, this suggested that the correct algebra to use for logic would be the common algebra of the ordinary numbers modified by restricting the symbols to the values 0 and 1.

He stated what, in this article, is called The Rule of 0 and 1 , that a law or argument held in logic iff after being translated into equational form it held in common algebra with this 0,1-restriction on the possible interpretations i. Boole would use this Rule to justify his main theorems Expansion, Reduction, Elimination , and for no other purpose.

Nonetheless it turns out that they do apply to his algebra of logic. In succeeding chapters he gave the Expansion Theorem, the new full-strength Elimination Theorem, an improved Reduction Theorem, and the use of division to solve an equation. After many examples and results for special cases of solving equations, Boole turned to the topic of the interpretability of a logical function.

Boole had already stated that every equation is interpretable by converting an equation into a collection of constituent equations. However terms need not be interpretable, e. Working with the modern notion of terms, one can recursively define the domain of interpretability of a term. In Chapter XIII Boole selected some well-known arguments of Clarke and Spinoza, on the nature of an eternal being, to put under the magnifying glass of his algebra of logic, starting with the comment LT , p.

In the final chapter on logic, chapter XV, Boole presented his analysis of the conversions and syllogisms of Aristotelian logic. He now considered this ancient logic to be a weak, fragmented attempt at a logical system. Briefly stated, Boole gave the reader a summary of traditional Aristotelian categorical logic, and analyzed some simple examples using ad hoc techniques with his algebra of logic.

Then he launched into proving a comprehensive result by applying his General Method to the pair of equations:. This led to three equations involving large algebraic expressions.

Boole omitted almost all details of his derivation, but summarized the results in terms of the established results of Aristotelian logic. Then he noted that the remaining categorical syllogisms are such that their premises can be put in the form:. In Laws of Thought , p. If my view be right, his system will come to be regarded as a most remarkable combination of truth and error. This would force every class symbol to denote the empty class.

It seems quite possible that Boole found the simplest way to construct an algebra of logic for classes that allowed one to use all the equations and equational arguments that were valid for the integers.

Each labeling of the universe creates a signed multi-set perhaps one should say signed multi-class consisting of those labeled elements where the label is non-zero. For multi-sets , whose labels are all non-negative, one can think of the label of an element as describing how many copies of the element are in the multi-set.

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