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ir exteriors which is what the left hand side of the law describes the second de morgan s law x y x y works the same way with the two diagrams interchanged the first complement law x x 0 says that the interior and exterior of the x circle have no overlap the second complement law x x 1 says that everything is either inside or outside the x circle digital logic gates edit 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 30 from left to right and or and not gates 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 the duality principle or de morgan s laws can be understood as asserting that complementing all three ports of an and gate converts it to an or gate and vice versa as shown in figure 4 below complementing both ports of an inverter however leaves the operation unchanged more generally one may complement any of the eight subsets of the three ports of either an and or or gate the resulting sixteen possibilities give rise to only eight boolean operations namely those with an odd number of 1s in their truth table there are eight such because the odd bit out can be either 0 or 1 and can go in any of four positions in the truth table there being sixteen binary boolean operations this must leave eight operations with an even number of 1s in their truth tables two of these are the constants 0 and 1 as binary operations that ignore both their inputs four are the operations that depend nontrivially on exactly one of their two inputs namely x y x and y and the remaining two are x y xor and its complement x y boolean algebras edit main article boolean algebra structure 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 concrete boolean algebras edit 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 5 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 however this exclusion conflicts with the preferred purely equational definition of boolean algebra there being no way to rule out the one element algebra using only equations 0 1 does not count being a negated 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 it can be seen that every field of subsets of x must contain the empty set and x hence no smaller example is possible other than the degenerate algebra obtained by taking x to be empty so as to make the empty set and x coincide example 3 the set of finite and cofinite sets of integers where a cofinite set is one omitting only finitely many integers this is clearly closed under complement and is closed under union because the union of a cofinite set with any set is cofinite while the union of two finite sets is finite intersection behaves like union with finite and cofinite interchanged this example is countably infinite because there are only countably many finite sets of integers example 4 for a less trivial example of the point made by example 2 consider a venn diagram formed by n closed curves partitioning the diagram into 2 n regions and let x be the infinite set of all points in the plane not on any curve but somewhere within the diagram the interior of each region is thus an infinite subset of x and every point in x is in exactly one region then the set of all 2 2 n possible unions of regions including the empty set obtained as the union of the empty set of regions and x obtained as the union of all 2 n regions is closed under union intersection and complement relative to x and therefore forms a concrete boolean algebra again there are finitely many subsets of an infinite set forming a concrete boolean algebra with example 2 arising as the case n 0 of no curves subsets as bit vectors edit a subset y of x can be identified with an indexed family of bits with index set x with the bit indexed by x x being 1 or 0 according to whether or not x y this is the so called characteristic function notion of a subset for example a 32 bit computer word consists of 32 bits indexed by the set 0 1 2 31 with 0 and 31 indexing the low and high order bits respectively for a smaller example if x a b c displaystyle x a b c where a b c are viewed as bit positions in that order from left to right the eight subsets c b b c a a c a b and a b c of x can be identified with the respective bit vectors 000 001 010 011 100 101 110 and 111 bit vectors indexed by the set of natural numbers are infinite sequences of bits while those indexed by the reals in the unit interval 0 1 are packed too densely to be able to write conventionally but nonetheless form well defined indexed families imagine coloring every point of the interval 0 1 either black or white independently the black points then form an arbitrary subset of 0 1 from this bit vector viewpoint a concrete boolean algebra can be defined equivalently as a nonempty set of bit vectors all of the same length more generally indexed by the same set and closed under the bit vector operations of bitwise and as in 1010 0110 0010 1010 0110 1110 and 1010 0101 the bit vector realizations of intersection union and complement respectively prototypical boolean algebra edit main article two element boolean algebra the set 0 1 and its boolean operations as treated above can be understood as the special case of bit vectors of length one which by the identification of bit vectors with subsets can also be understood as the two subsets of a one element set this is called the prototypical boolean algebra justified by the following observation the laws satisfied by all nondegenerate concrete boolean algebras coincide with those satisfied by the prototypical boolean algebra this observation is proved as follows certainly any law satisfied by all concrete boolean algebras is satisfied by the prototypical one since it is concrete conversely any law that fails for some concrete boolean algebra must have failed at a particular bit position in which case that position by itself furnishes a one bit counterexample to that law nondegeneracy ensures the existence of at least one bit position because there is only one empty bit vector the final goal of the next section can be understood as eliminating concrete from the above observation that goal is reached via the stronger observation that up to isomorphism all boolean algebras are concrete boolean algebras the definition edit the boolean algebras so far have all been concrete consisting of bit vectors or equivalently of subsets of some set such a boolean algebra consists of a set and operations on that set which can be shown to satisfy the laws of boolean algebra instead of showing that the boolean laws are satisfied we can instead postulate a set x two binary operations on x and one unary operation and require that those operations satisfy the laws of boolean algebra the elements of x need not be bit vectors or subsets but can be anything at all this leads to the more general abstract definition a boolean algebra is any set with binary operations and and a unary operation thereon satisfying the boolean laws 31 for the purposes of this definition it is irrelevant how the operations came to satisfy the laws whether by fiat or proof all concrete boolean algebras satisfy the laws by proof rather than fiat whence every concrete boolean algebra is a boolean algebra according to our definitions this axiomatic definition of a boolean algebra as a set and certain operations satisfying certain laws or axioms by fiat is entirely analogous to the abstract definitions of group ring field etc characteristic of modern or abstract algebra given any complete axiomatization of boolean algebra such as the axioms for a complemented distributive lattice a sufficient condition for an algebraic structure of this kind to satisfy all the boolean laws is that it satisfy just those axioms the following is therefore an equivalent definition a boolean algebra is a complemented distributive lattice the section on axiomatization lists other axiomatizations any of which can be made the basis of an equivalent definition representable boolean algebras edit although every concrete boolean algebra is a boolean algebra not every boolean algebra need be concrete let n be a square free positive integer one not divisible by the square of an integer for example 30 but not 12 the operations of greatest common divisor least common multiple and division into n that is x n x can be shown to satisfy all the boolean laws when their arguments range over the positive divisors of n hence those divisors form a boolean algebra these divisors are not subsets of a set making the divisors of n a boolean algebra that is not concrete according to our definitions however if each divisor of n is represented by the set of its prime factors this nonconcrete boolean algebra is isomorphic to the concrete boolean algebra consisting of all sets of prime factors of n with union corresponding to least common multiple intersection to greatest common divisor and complement to division into n so this example while not technically concrete is at least morally concrete via this representation called an isomorphism this example is an instance of the following notion a boolean algebra is called representable when it is isomorphic to a concrete boolean algebra the next question is answered positively as follows every boolean algebra is representable that is up to isomorphism abstract and concrete boolean algebras are the same thing this result depends on the boolean prime ideal theorem a choice principle slightly weaker than the axiom of choice this strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability the laws satisfied by all boolean algebras coincide with those satisfied by the prototypical boolean algebra it is weaker in the sense that it does not of itself imply representability boolean algebras are special here for example a relation algebra is a boolean algebra with additional structure but it is not the case that every relation algebra is representable in the sense appropriate to relation algebras axiomatizing boolean algebra edit main articles axiomatization of boolean algebras and boolean algebras canonically defined the above definition of an abstract boolean algebra as a set together with operations satisfying the boolean laws raises the question of what those laws are a simplistic answer is all boolean laws which can be defined as all equations that hold for the boolean algebra of 0 and 1 however since there are infinitely many such laws this is not a satisfactory answer in practice leading to the question of it suffices to require only finitely many laws to hold in the case of boolean algebras the answer is yes the finitely many equations listed above are sufficient thus boolean algebra is said to be finitely axiomatizable or finitely based moreover the number of equations needed can be further reduced to begin with some of the above laws are implied by some of the others a sufficient subset of the above laws consists of the pairs of associativity commutativity and absorption laws distributivity of over or the other distributivity law one suffices and the two complement laws in fact this is the traditional axiomatization of boolean algebra as a complemented distributive lattice by introducing additional laws not listed above it becomes possible to shorten the list of needed equations yet further for instance with the vertical bar representing the sheffer stroke operation the single axiom a b c a a c a c displaystyle a mid b mid c mid a mid a mid c mid a c is sufficient to completely axiomatize boolean algebra it is also possible to find longer single axioms using more conventional operations see minimal axioms for boolean algebra 32 propositional logic edit main article propositional calculus propositional logic is a logical system that is intimately connected to boolean algebra 5 many syntactic concepts of boolean algebra carry over to propositional logic with only minor changes in notation and terminology while the semantics of propositional logic are defined via boolean algebras in a way that the tautologies theorems of propositional logic correspond to equational theorems of boolean algebra syntactically every boolean term corresponds to a propositional formula of propositional logic in this translation between boolean algebra and propositional logic boolean variables x y become propositional variables or atoms p q boolean terms such as x y become propositional formulas p q 0 becomes false or and 1 becomes true or it is convenient when referring to generic propositions to use greek letters φ ψ as metavariables variables outside the language of propositional calculus used when talking about propositional calculus to denote propositions the semantics of propositional logic rely on truth assignments the essential idea of a truth assignment is that the propositional variables a...
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