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 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. Nonmonotonicity enters via complement ¬ as follows.^{ [4] }
The complement operation is defined by the following two laws.
All properties of negation including the laws below follow from the above two laws alone.^{ [4] }
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)
But whereas ordinary algebra satisfies the two laws
Boolean algebra satisfies De Morgan's laws:
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 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. 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 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.
Principle: If {X, R} is a poset, then {X, R(inverse)} is also a poset.
There is nothing magical about the choice of symbols for the values of Boolean algebra. We could rename 0 and 1 to say α and β, and as long as we did so consistently throughout it would still be Boolean algebra, albeit with some obvious cosmetic differences.
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. However it would not be identical to our original Boolean algebra because now we find ∨ behaving the way ∧ used to do and vice versa. 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. We might notice that the columns for x∧y and x∨y in the truth tables had changed places, but that switch is immaterial.
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. Thus 0 and 1 are dual, and ∧ and ∨ are dual. 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. A more complicated example of a self-dual operation is (x∧y) ∨ (y∧z) ∨ (z∧x). There is no self-dual binary operation that depends on both its arguments. A composition of self-dual operations is a self-dual operation. For example, if f(x, y, z) = (x∧y) ∨ (y∧z) ∨ (z∧x), then f(f(x, y, z), x, t) is a self-dual operation of four arguments x,y,z,t.
The principle of duality can be explained from a group theory perspective by the 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. Walter Gottschalk remarked that consequently a more appropriate name for the phenomenon would be the principle (or square) of quaternality.^{ [15] }
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).
The three Venn diagrams in the figure below represent respectively conjunction x∧y, disjunction x∨y, and complement ¬x.
For conjunction, the region inside both circles is shaded to indicate that x∧y is 1 when both variables are 1. The other regions are left unshaded to indicate that x∧y is 0 for the other three combinations.
The second diagram represents disjunction x∨y by shading those regions that lie inside either or both circles. The third diagram represents complement ¬x by shading the region not inside the circle.
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 commutativity laws for ∧ and ∨ can be seen from the symmetry of the diagrams: a binary operation that was not commutative would not have a symmetric diagram because interchanging x and y would have the effect of reflecting the diagram horizontally and any failure of commutativity would then appear as a failure of symmetry.
Idempotence of ∧ and ∨ can be visualized by sliding the two circles together and noting that the shaded area then becomes the whole circle, for both ∧ and ∨.
To see the first absorption law, x∧(x∨y) = x, start with the diagram in the middle for x∨y and note that the portion of the shaded area in common with the x circle is the whole of the x circle. For the second absorption law, x∨(x∧y) = x, start with the left diagram for x∧y and note that shading the whole of the x circle results in just the x circle being shaded, since the previous shading was inside the x circle.
The double negation law can be seen by complementing the shading in the third diagram for ¬x, which shades the x circle.
To visualize the first De Morgan's law, (¬x)∧(¬y) = ¬(x∨y), start with the middle diagram for x∨y and complement its shading so that only the region outside both circles is shaded, which is what the right hand side of the law describes. The result is the same as if we shaded that region which is both outside the x circle and outside the y circle, i.e. the conjunction of their 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 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.^{ [17] }
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 1's 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 1's 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.
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.^{ [4] }
(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. 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.
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^{2n} 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 we have finitely many subsets of an infinite set forming a concrete Boolean algebra, with Example 2 arising as the case n = 0 of no curves.
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} 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.
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. We call this the prototypical Boolean algebra, justified by the following observation.
This observation is easily 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. We shall however reach that goal via the surprisingly stronger observation that, up to isomorphism, all Boolean algebras are concrete.
The Boolean algebras we have seen 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.
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.
The section on axiomatization lists other axiomatizations, any of which can be made the basis of an equivalent definition.
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 we represent each divisor of n by the set of its prime factors, we find that 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.
The obvious next question is answered positively as follows.
That is, up to isomorphism, abstract and concrete Boolean algebras are the same thing. This quite nontrivial result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice, and is treated in more detail in the article Stone's representation theorem for Boolean algebras. This strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability.
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.
The above definition of an abstract Boolean algebra as a set and operations satisfying "the" Boolean laws raises the question, what are those laws? A simple-minded answer is "all Boolean laws," which can be defined as all equations that hold for the Boolean algebra of 0 and 1. Since there are infinitely many such laws this is not a terribly satisfactory answer in practice, leading to the next question: does it suffice to require only finitely many laws to hold?
In the case of Boolean algebras the answer is yes. In particular the finitely many equations we have listed above suffice. We say that Boolean algebra is finitely axiomatizable or finitely based.
Can this list be made shorter yet? Again the answer is yes. 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 yet further. In 1933, Edward Huntington showed that if the basic operations are taken to be x∨y and ¬x, with x∧y considered a derived operation (e.g. via De Morgan's law in the form x∧y = ¬(¬x∨¬y)), then the equation ¬(¬x∨¬y)∨¬(¬x∨y) = x along with the two equations expressing associativity and commutativity of ∨ completely axiomatized Boolean algebra. When the only basic operation is the binary NAND operation ¬(x∧y), Stephen Wolfram has proposed in his book A New Kind of Science the single axiom ((xy)z)(x((xz)x)) = z as a one-equation axiomatization of Boolean algebra, where for convenience here xy denotes the NAND rather than the AND of x and y.
Propositional logic is a logical system that is intimately connected to Boolean algebra.^{ [4] } 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 T. 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 assignment s. The essential idea of a truth assignment is that the propositional variables are mapped to elements of a fixed Boolean algebra, and then the truth value of a propositional formula using these letters is the element of the Boolean algebra that is obtained by computing the value of the Boolean term corresponding to the formula. In classical semantics, only the two-element Boolean algebra is used, while in Boolean-valued semantics arbitrary Boolean algebras are considered. A tautology is a propositional formula that is assigned truth value 1 by every truth assignment of its propositional variables to an arbitrary Boolean algebra (or, equivalently, every truth assignment to the two element Boolean algebra).
These semantics permit a translation between tautologies of propositional logic and equational theorems of Boolean algebra. Every tautology Φ of propositional logic can be expressed as the Boolean equation Φ = 1, which will be a theorem of Boolean algebra. Conversely every theorem Φ = Ψ of Boolean algebra corresponds to the tautologies (Φ∨¬Ψ) ∧ (¬Φ∨Ψ) and (Φ∧Ψ) ∨ (¬Φ∧¬Ψ). If → is in the language these last tautologies can also be written as (Φ→Ψ) ∧ (Ψ→Φ), or as two separate theorems Φ→Ψ and Ψ→Φ; if ≡ is available then the single tautology Φ ≡ Ψ can be used.
One motivating application of propositional calculus is the analysis of propositions and deductive arguments in natural language. Whereas the proposition "if x = 3 then x+1 = 4" depends on the meanings of such symbols as + and 1, the proposition "if x = 3 then x = 3" does not; it is true merely by virtue of its structure, and remains true whether "x = 3" is replaced by "x = 4" or "the moon is made of green cheese." The generic or abstract form of this tautology is "if P then P", or in the language of Boolean algebra, "P → P".
Replacing P by x = 3 or any other proposition is called instantiation of P by that proposition. The result of instantiating P in an abstract proposition is called an instance of the proposition. Thus "x = 3 → x = 3" is a tautology by virtue of being an instance of the abstract tautology "P → P". All occurrences of the instantiated variable must be instantiated with the same proposition, to avoid such nonsense as P → x = 3 or x = 3 → x = 4.
Propositional calculus restricts attention to abstract propositions, those built up from propositional variables using Boolean operations. Instantiation is still possible within propositional calculus, but only by instantiating propositional variables by abstract propositions, such as instantiating Q by Q→P in P→(Q→P) to yield the instance P→((Q→P)→P).
(The availability of instantiation as part of the machinery of propositional calculus avoids the need for metavariables within the language of propositional calculus, since ordinary propositional variables can be considered within the language to denote arbitrary propositions. The metavariables themselves are outside the reach of instantiation, not being part of the language of propositional calculus but rather part of the same language for talking about it that this sentence is written in, where we need to be able to distinguish propositional variables and their instantiations as being distinct syntactic entities.)
An axiomatization of propositional calculus is a set of tautologies called axioms and one or more inference rules for producing new tautologies from old. A proof in an axiom system A is a finite nonempty sequence of propositions each of which is either an instance of an axiom of A or follows by some rule of A from propositions appearing earlier in the proof (thereby disallowing circular reasoning). The last proposition is the theorem proved by the proof. Every nonempty initial segment of a proof is itself a proof, whence every proposition in a proof is itself a theorem. An axiomatization is sound when every theorem is a tautology, and complete when every tautology is a theorem.^{ [19] }
Propositional calculus is commonly organized as a Hilbert system, whose operations are just those of Boolean algebra and whose theorems are Boolean tautologies, those Boolean terms equal to the Boolean constant 1. Another form is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions called sequents, such as A∨B, A∧C,... A, B→C,.... The two halves of a sequent are called the antecedent and the succedent respectively. The customary metavariable denoting an antecedent or part thereof is Γ, and for a succedent Δ; thus Γ,A Δ would denote a sequent whose succedent is a list Δ and whose antecedent is a list Γ with an additional proposition A appended after it. The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by the antecedent.
Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Boolean algebra, the former is a binary relation which either holds or does not hold. In this sense entailment is an external form of implication, meaning external to the Boolean algebra, thinking of the reader of the sequent as also being external and interpreting and comparing antecedents and succedents in some Boolean algebra. The natural interpretation of is as ≤ in the partial order of the Boolean algebra defined by x ≤ y just when x∨y = y. This ability to mix external implication and internal implication → in the one logic is among the essential differences between sequent calculus and propositional calculus.^{ [20] }
Boolean algebra as the calculus of two values is fundamental to computer circuits, computer programming, and mathematical logic, and is also used in other areas of mathematics such as set theory and statistics.^{ [4] }
In the early 20th century, several electrical engineers intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits. Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits .
Today, all modern general purpose computers perform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic. They achieve this in various ways: as voltages on wires in high-speed circuits and capacitive storage devices, as orientations of a magnetic domain in ferromagnetic storage devices, as holes in punched card s or paper tape, and so on. (Some early computers used decimal circuits or mechanisms instead of two-valued logic circuits.)
Of course, it is possible to code more than two symbols in any given medium. For example, one might use respectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of different sizes in a punched card. In practice, the tight constraints of high speed, small size, and low power combine to make noise a major factor. This makes it hard to distinguish between symbols when there are several possible symbols that could occur at a single site. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per wire, high and low.
Computers use two-value Boolean circuits for the above reasons. The most common computer architectures use ordered sequences of Boolean values, called bits, of 32 or 64 values, e.g. 01101000110101100101010101001011. When programming in machine code, assembly language, and certain other programming languages, programmers work with the low-level digital structure of the data registers. These registers operate on voltages, where zero volts represents Boolean 0, and a reference voltage (often +5V, +3.3V, +1.8V) represents Boolean 1. Such languages support both numeric operations and logical operations. In this context, "numeric" means that the computer treats sequences of bits as binary numbers (base two numbers) and executes arithmetic operations like add, subtract, multiply, or divide. "Logical" refers to the Boolean logical operations of disjunction, conjunction, and negation between two sequences of bits, in which each bit in one sequence is simply compared to its counterpart in the other sequence. Programmers therefore have the option of working in and applying the rules of either numeric algebra or Boolean algebra as needed. A core differentiating feature between these families of operations is the existence of the carry operation in the first but not the second.
Other areas where two values is a good choice are the law and mathematics. In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In more focused situations such as a court of law or theorem-based mathematics however it is deemed advantageous to frame questions so as to admit a simple yes-or-no answer—is the defendant guilty or not guilty, is the proposition true or false—and to disallow any other answer. However much of a straitjacket this might prove in practice for the respondent, the principle of the simple yes-no question has become a central feature of both judicial and mathematical logic, making two-valued logic deserving of organization and study in its own right.
A central concept of set theory is membership. Now an organization may permit multiple degrees of membership, such as novice, associate, and full. With sets however an element is either in or out. The candidates for membership in a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just as each wire is either high or low.
Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory.
Two-valued logic can be extended to multi-valued logic, notably by replacing the Boolean domain {0, 1} with the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x, conjunction (AND) is replaced with multiplication (), and disjunction (OR) is defined via De Morgan's law. Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
The original application for Boolean operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.
Natural languages such as English have words for several Boolean operations, in particular conjunction (and), disjunction (or), negation (not), and implication (implies). But not is synonymous with and not. When used to combine situational assertions such as "the block is on the table" and "cats drink milk," which naively are either true or false, the meanings of these logical connectives often have the meaning of their logical counterparts. However, with descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failure of commutativity, for example the conjunction of "Jim opened the door" with "Jim walked through the door" in that order is not equivalent to their conjunction in the other order, since and usually means and then in such cases. Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverse order. Conjunctive commands about behavior are like behavioral assertions, as in get dressed and go to school. Disjunctive commands such love me or leave me or fish or cut bait tend to be asymmetric via the implication that one alternative is less preferable. Conjoined nouns such as tea and milk generally describe aggregation as with set union while tea or milk is a choice. However context can reverse these senses, as in your choices are coffee and tea which usually means the same as your choices are coffee or tea (alternatives). Double negation as in "I don't not like milk" rarely means literally "I do like milk" but rather conveys some sort of hedging, as though to imply that there is a third possibility. "Not not P" can be loosely interpreted as "surely P", and although P necessarily implies "not not P" the converse is suspect in English, much as with intuitionistic logic. In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.
Boolean operations are used in digital logic to combine the bits carried on individual wires, thereby interpreting them over {0,1}. When a vector of n identical binary gates are used to combine two bit vectors each of n bits, the individual bit operations can be understood collectively as a single operation on values from a Boolean algebra with 2^{n} elements.
Naive set theory interprets Boolean operations as acting on subsets of a given set X. As we saw earlier this behavior exactly parallels the coordinate-wise combinations of bit vectors, with the union of two sets corresponding to the disjunction of two bit vectors and so on.
The 256-element free Boolean algebra on three generators is deployed in computer displays based on raster graphics, which use bit blit to manipulate whole regions consisting of pixels, relying on Boolean operations to specify how the source region should be combined with the destination, typically with the help of a third region called the mask. Modern video cards offer all 2^{23} = 256 ternary operations for this purpose, with the choice of operation being a one-byte (8-bit) parameter. The constants SRC = 0xaa or 10101010, DST = 0xcc or 11001100, and MSK = 0xf0 or 11110000 allow Boolean operations such as (SRC^DST)&MSK (meaning XOR the source and destination and then AND the result with the mask) to be written directly as a constant denoting a byte calculated at compile time, 0x60 in the (SRC^DST)&MSK example, 0x66 if just SRC^DST, etc. At run time the video card interprets the byte as the raster operation indicated by the original expression in a uniform way that requires remarkably little hardware and which takes time completely independent of the complexity of the expression.
Solid modeling systems for computer aided design offer a variety of methods for building objects from other objects, combination by Boolean operations being one of them. In this method the space in which objects exist is understood as a set S of voxels (the three-dimensional analogue of pixels in two-dimensional graphics) and shapes are defined as subsets of S, allowing objects to be combined as sets via union, intersection, etc. One obvious use is in building a complex shape from simple shapes simply as the union of the latter. Another use is in sculpting understood as removal of material: any grinding, milling, routing, or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer with the Boolean operation x ∧ ¬y or x − y, which in set theory is set difference, remove the elements of y from those of x. Thus given two shapes one to be machined and the other the material to be removed, the result of machining the former to remove the latter is described simply as their set difference.
Search engine queries also employ Boolean logic. For this application, each web page on the Internet may be considered to be an "element" of a "set". The following examples use a syntax previously supported by Google.^{ [21] }
"Search term 1" "Search term 2"
"Search term 1" OR "Search term 2"
"Search term 1" −"Search term 2"
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.
In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is typically written as ∧ or ⋅ .
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand or the alternative denial, since it says in effect that at least one of its operands is false. In digital electronics, it corresponds to the NAND gate. It is named after Henry M. Sheffer and written as ↑ or as |. In Bocheński notation it can be written as Dpq.
In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctive clauses; it can also be described as an OR of ANDs, a sum of products, or a cluster concept. As a normal form, it is useful in automated theorem proving.
In logic, negation, also called the logical complement, is an operation that takes a proposition to another proposition "not ", written , which is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.
In mathematics, a Heyting algebra is a bounded lattice equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Equivalently a Heyting algebra is a residuated lattice whose monoid operation is ∧; yet another definition is as a posetal cartesian closed category with all finite sums. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic.
In logic, a truth function is a function that accepts truth values as input and produces a truth value as output, i.e., the input and output are all truth values. The typical example is in propositional logic, wherein a compound statement is constructed by one or two statements connected by a logical connective; if the truth value of the compound statement is determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and the logical connective is said to be truth functional.
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation , a unary operation , and the constant , satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the many-valued logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras.
In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that:
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2^{X²} of all binary relations on a set X, that is, subsets of the cartesian square X^{2}, with R•S interpreted as the usual composition of binary relations R and S, and with the converse of R as the converse relation.
In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression. A well-known complete set of connectives is { AND, NOT }, consisting of binary conjunction and negation. Each of the singleton sets { NAND } and { NOR } is functionally complete.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
Boolean algebra is a mathematically rich branch of abstract algebra. Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under zero or more operations satisfying certain equations.
In mathematics and philosophy, Łukasiewicz logic is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic; it was later generalized to n-valued as well as infinitely-many-valued (ℵ_{0}-valued) variants, both propositional and first-order. The ℵ_{0}-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz–Tarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics.
T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning.
Łukasiewicz–Moisil algebras were introduced in the 1940s by Grigore Moisil in the hope of giving algebraic semantics for the n-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz logic. A faithful model for the ℵ_{0}-valued (infinitely-many-valued) Łukasiewicz–Tarski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated (finite) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV_{n}-algebras. MV_{n}-algebras are a subclass of LM_{n}-algebras, and the inclusion is strict for n ≥ 5. In 1982 Roberto Cignoli published some additional constraints that added to LM_{n}-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.
Mano, Morris; Ciletti, Michael D. (2013). Digital Design. Pearson. ISBN 978-0-13-277420-8.
Historical perspective
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