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harles sanders peirce gave the title a boolian sic algebra with one constant to the first chapter of his the simplest mathematics in 1880 4 boolean algebra has been fundamental in the development of digital electronics and is provided for in all modern programming languages it is also used in set theory and statistics 5 history edit a precursor of boolean algebra was gottfried wilhelm leibniz s algebra of concepts the usage of binary in relation to the i ching was central to leibniz s characteristica universalis it eventually created the foundations of algebra of concepts 6 leibniz s algebra of concepts is deductively equivalent to the boolean algebra of sets 7 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 8 in an abstract setting boolean algebra was perfected in the late 19th century by jevons schröder huntington and others until it reached the modern conception of an abstract mathematical structure 8 for example the empirical observation that one can manipulate expressions in the algebra of sets by translating them into expressions in boole s algebra is explained in modern terms by saying that the algebra of sets is a boolean algebra note the indefinite article in fact m h stone proved in 1936 that every boolean algebra is isomorphic to a field of sets 9 10 in the 1930s while studying switching circuits claude shannon observed that one could also apply the rules of boole s algebra in this setting 11 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 modern circuit engineering settings there is little need to consider other boolean algebras thus switching algebra and boolean algebra are often used interchangeably 12 13 14 efficient implementation of boolean functions is a fundamental problem in the design of combinational logic circuits modern electronic design automation tools for very large scale integration 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 15 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 16 17 18 boolean algebra is not sufficient to capture logic formulas using quantifiers like those from first order logic 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 8 the problem of determining whether the variables of a given boolean propositional formula can be assigned in such a way as to make the formula evaluate to true is called the boolean satisfiability problem sat and is of importance to theoretical computer science being the first problem shown to be np complete the closely related model of computation known as a boolean circuit relates time complexity of an algorithm to circuit complexity values edit whereas expressions denote mainly numbers in elementary algebra in boolean algebra they denote the truth values false and true these values are represented with the bits 0 and 1 they do not behave like the integers 0 and 1 for which 1 1 2 but may be identified with the elements of the two element field gf 2 that is integer arithmetic modulo 2 for which 1 1 0 addition and multiplication then play the boolean roles of xor exclusive or and and conjunction respectively with disjunction x y inclusive or definable as x y xy and negation x as 1 x in gf 2 may be replaced by since they denote the same operation however this way of writing boolean operations allows applying the usual arithmetic operations of integers this may be useful when using a programming language in which gf 2 is not implemented boolean algebra also deals with functions which have their values in the set 0 1 a sequence of bits is a commonly used example of such a function another common example is the totality of subsets of a set e to a subset f of e one can define the indicator function that takes the value 1 on f and 0 outside f the most general example is the set elements of a boolean algebra with all of the foregoing being instances thereof as with elementary algebra the purely equational part of the theory may be developed without considering explicit values for the variables 19 operations edit further information truth table this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed find sources boolean algebra news newspapers books scholar jstor april 2019 learn how and when to remove this message basic operations edit while elementary algebra has four operations addition subtraction multiplication and division the boolean algebra has only three basic operations conjunction disjunction and negation expressed with the corresponding binary operators and displaystyle land and or displaystyle lor and the unary operator not displaystyle neg collectively referred to as boolean operators 20 variables in boolean algebra that store the logical value of 0 and 1 are called the boolean variables they are used to store either true or false values 21 the basic operations on boolean variables x and y are defined as follows logical operation operator notation alternative notations definition conjunction and x y x and y k xy x y 1 if x y 1 x y 0 otherwise disjunction or x y x or y a xy x y 0 if x y 0 x y 1 otherwise negation not x not x n x x̅ x x x 0 if x 1 x 1 if x 0 alternatively the values of x y x y and x can be expressed by tabulating their values with truth tables as follows 22 x displaystyle x y displaystyle y x y displaystyle x wedge y x y displaystyle x vee y 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 x displaystyle x x displaystyle neg x 0 1 1 0 when used in expressions the operators are applied according to the precedence rules as with elementary algebra expressions in parentheses are evaluated first following the precedence rules 23 if the truth values 0 and 1 are interpreted as integers these operations may be expressed with the ordinary operations of arithmetic where x y uses addition and xy uses multiplication or by the minimum maximum functions x y x y min x y x y x y x y x y 1 x max x y x 1 x displaystyle begin aligned x wedge y xy min x y x vee y x y xy x y 1 x max x y neg x 1 x end aligned one might consider that only negation and one of the two other operations are basic because of the following identities that allow one to define conjunction in terms of negation and the disjunction and vice versa de morgan s laws 24 x y x y x y x y displaystyle begin aligned x wedge y neg neg x vee neg y x vee y neg neg x wedge neg y end aligned secondary operations edit operations composed from the basic operations include among others the following material conditional x y x y textstyle x rightarrow y neg x vee y material biconditional x y x y x y x y x y textstyle x leftrightarrow y x land y lor neg x land neg y x lor neg y land neg x lor y exclusive or xor x y x y x y x y x y x y x y x y textstyle x oplus y neg x leftrightarrow y x vee y wedge neg x wedge y x vee y wedge neg x vee neg y x wedge neg y vee neg x wedge y these definitions give rise to the following truth tables giving the values of these operations for all four possible inputs secondary operations table 1 x displaystyle x y displaystyle y x y displaystyle x rightarrow y x y displaystyle x oplus y x y x y displaystyle x leftrightarrow y x equiv y 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 material conditional the first operation x y or c xy is called material implication if x is true then the result of expression x y is taken to be that of y e g if x is true and y is false then x y is also false but if x is false then the value of y can be ignored however the operation must return some boolean value and there are only two choices so by definition x y is true when x is false relevance logic rejects this definition by viewing an implication with a false premise as something other than either true or false exclusive or xor the second operation x y or j xy is called exclusive or often abbreviated as xor to distinguish it from disjunction as the inclusive kind it excludes the possibility of both x and y being true e g see table if both are true then result is false defined in terms of arithmetic it is addition where mod 2 is 1 1 0 logical equivalence the third operation the complement of exclusive or is equivalence or boolean equality x y or e xy is true just when x and y have the same value hence x y as its complement can be understood as x y being true just when x and y are different thus its counterpart in arithmetic mod 2 is x y equivalence s counterpart in arithmetic mod 2 is x y 1 laws edit a law of boolean algebra is an identity such as x y z x y z between two boolean terms where a boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations and the concept can be extended to terms involving other boolean operations such as and but such extensions are unnecessary for the purposes to which the laws are put such purposes include the definition of a boolean algebra as any model of the boolean laws and as a means for deriving new laws from old as in the derivation of x y z x z y from y z z y as treated in axiomatizing boolean algebra monotone laws edit boolean algebra satisfies many of the same laws as ordinary algebra when one matches up with addition and with multiplication in particular the following laws are common to both kinds of algebra 25 26 associativity of x y z displaystyle x vee y vee z x y z displaystyle x vee y vee z associativity of x y z displaystyle x wedge y wedge z x y z displaystyle x wedge y wedge z commutativity of x y displaystyle x vee y y x displaystyle y vee x commutativity of x y displaystyle x wedge y y x displaystyle y wedge x distributivity of over x y z displaystyle x wedge y vee z x y x z displaystyle x wedge y vee x wedge z identity for x 0 displaystyle x vee 0 x displaystyle x identity for x 1 displaystyle x wedge 1 x displaystyle x annihilator for x 0 displaystyle x wedge 0 0 displaystyle 0 the following laws hold in boolean algebra but not in ordinary algebra annihilator for x 1 displaystyle x vee 1 1 displaystyle 1 idempotence of x x displaystyle x vee x x displaystyle x idempotence of x x displaystyle x wedge x x displaystyle x absorption 1 x x y displaystyle x wedge x vee y x displaystyle x absorption 2 x x y displaystyle x vee x wedge y x displaystyle x distributivity of over x y z displaystyle x vee y wedge z x y x z displaystyle x vee y wedge x vee z taking x 2 in the third law above shows that it is not an ordinary algebra law since 2 2 4 the remaining five laws can be falsified in ordinary algebra by taking all variables to be 1 for example in absorption law 1 the left hand side would be 1 1 1 2 while the right hand side would be 1 and so on all of the laws treated thus 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 and vice versa operations with this property are said to be monotone thus the axioms thus far have all been for monotonic boolean logic nonmonotonicity enters via complement as follows 5 nonmonotone laws edit the complement operation is defined by the following two laws complementation 1 x x 0 complementation 2 x x 1 displaystyle begin aligned text complementation 1 x wedge neg x 0 text complementation 2 x vee neg x 1 end aligned all properties of negation including the laws below follow from the above two laws alone 5 in both ordinary and boolean algebra negation works by exchanging pairs of elements hence in both algebras it satisfies the double negation law also called involution law double negation x x displaystyle begin aligned text double negation neg neg x x end aligned but whereas ordinary algebra satisfies the two laws x y x y x y x y displaystyle begin aligned x y xy x y x y end aligned boolean algebra satisfies de morgan s laws de morgan 1 x y x y de morgan 2 x y x y displaystyle begin aligned text de morgan 1 neg x wedge neg y neg x vee y text de morgan 2 neg x vee neg y neg x wedge y end aligned completeness edit the laws listed above define boolean algebra in the sense that they 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 boolean algebras 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 in contrast in a list of some but not all of the same laws there could have been boolean laws that did not follow from those on the 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 attention was not paid as to whether some of the axioms followed from others but there was simply a choice to stop when enough laws had been noticed treated further in axiomatizing boolean algebra 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 27 28 all these definitions of boolean algebra can be shown to be equivalent duality principle edit main article conjunction disjunction duality principle if x r is a partially ordered set then x r inverse is also a partially ordered set there is nothing special about the choice of symbols for the values of boolean algebra 0 and 1 could be renamed to α and β and as long as it was done consistently throughout it would still be boolean algebra albeit with some obvious cosmetic differences but suppose 0 and 1 were renamed 1 and 0 respectively then it would still be boolean algebra and moreover operating on the same values however it would not be identical to our original boolean algebra because now behaves the way used to do and vice versa so there are still some cosmetic differences to show that the notation has been changed d...
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