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39-550: Boolean-valued usually refers to: in most applied fields: something taking one of two values (example: True or False, On or Off, 1 or 0) referring to two-element Boolean algebra (the Boolean domain ), e.g. Boolean-valued function or Boolean data type in mathematics: something taking values over an arbitrary, abstract Boolean algebra , for example Boolean-valued model See also [ edit ] Boolean algebra further explains

78-421: A + b = b + a and ab = ba are commutative laws . Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, the possible moves of an object in three-dimensional space can be combined by performing a first move of the object, and then a second move from its new position. Such moves, formally called rigid motions , obey

117-479: A decision procedure . Logicians refer to this fact as " 2 is decidable ". All known decision procedures require a number of steps that is an exponential function of the number of variables N appearing in the equation to be verified. Whether there exists a decision procedure whose steps are a polynomial function of N falls under the P ;= NP conjecture. The above metatheorem does not hold if we consider

156-445: A (countable) set of variables x , y , z , etc. the term algebra is the collection of all possible terms involving m , i , e and the variables; so for example, m ( i ( x ), m ( x , m ( y , e ))) would be an element of the term algebra. One of the axioms defining a group is the identity m ( x , i ( x )) = e ; another is m ( x , e ) = x . The axioms can be represented as trees . These equations induce equivalence classes on

195-424: A class of algebras are identities, then this class is a variety (not to be confused with algebraic varieties of algebraic geometry ). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe . Identities contain no connectives , existentially quantified variables , or relations of any kind other than

234-412: A complete list, but include the most common structures taught in undergraduate courses. An axiom of an algebraic structure often has the form of an identity , that is, an equation such that the two sides of the equals sign are expressions that involve operations of the algebraic structure and variables . If the variables in the identity are replaced by arbitrary elements of the algebraic structure,

273-404: A finite set of identities (known as axioms ) that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field , and an operation called scalar multiplication between elements of the field (called scalars ), and elements of

312-417: A function can be used to drive all complements down to the individual variables. A powerful and nontrivial metatheorem states that any identity of 2 holds for all Boolean algebras. Conversely, an identity that holds for an arbitrary nontrivial Boolean algebra also holds in 2 . Hence all identities of Boolean algebra are captured by 2 . This theorem is useful because any equation in 2 can be verified by

351-464: A set of equational identities (the axioms), one may consider their symmetric, transitive closure E . The quotient algebra T / E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m , taking two arguments, and the inverse operator i , taking one argument, and the identity element e , a constant, which may be considered an operator that takes zero arguments. Given

390-406: A typical axiom is inversion in fields . This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a variety in the sense of universal algebra .) It can be stated: "Every nonzero element of a field is invertible ;" or, equivalently: the structure has a unary operation inv such that The operation inv can be viewed either as a partial operation that

429-408: A variety. Here are some of the most common existential axioms. The axioms of an algebraic structure can be any first-order formula , that is a formula involving logical connectives (such as "and" , "or" and "not" ), and logical quantifiers ( ∀ , ∃ {\displaystyle \forall ,\exists } ) that apply to elements (not to subsets) of the structure. Such

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468-555: Is a ⟨ B , + , {\displaystyle \langle B,+,} ∙ , . . ¯ , 1 , 0 ⟩ {\displaystyle ,{\overline {..}},1,0\rangle } algebra of type ⟨ 2 , 2 , 1 , 0 , 0 ⟩ {\displaystyle \langle 2,2,1,0,0\rangle } . Either one-to-one correspondence between {0,1} and { True , False } yields classical bivalent logic in equational form, with complementation read as NOT . If 1

507-437: Is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure. There are various concepts in category theory that try to capture the algebraic character of a context, for instance In a slight abuse of notation , the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example,

546-555: Is a k - tuple of variables. Choosing a specific value of y for each value of X defines a function φ : X ↦ y , {\displaystyle \varphi :X\mapsto y,} which can be viewed as an operation of arity k , and the axiom becomes the identity f ( X , φ ( X ) ) = g ( X , φ ( X ) ) . {\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).} The introduction of such auxiliary operation complicates slightly

585-581: Is also that of Quine 's Boolean term schemata . Letting ( X ) denote the complement of X and "()" denote either 0 or 1 yields the syntax of the primary algebra of G. Spencer-Brown 's Laws of Form . A basis for 2 is a set of equations, called axioms , from which all of the above equations (and more) can be derived. There are many known bases for all Boolean algebras and hence for 2 . An elegant basis notated using only concatenation and overbar is: Where concatenation = OR, 1 = true, and 0 = false, or concatenation = AND, 1 = false, and 0 = true. (overbar

624-470: Is called an algebra ; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure that is a vector space over a field or a module over a commutative ring . The collection of all structures of a given type (same operations and same laws) is called a variety in universal algebra; this term is also used with a completely different meaning in algebraic geometry , as an abbreviation of algebraic variety . In category theory,

663-533: Is different from Wikidata All set index articles Two-element Boolean algebra B is a partially ordered set and the elements of B are also its bounds . An operation of arity n is a mapping from B to B . Boolean algebra consists of two binary operations and unary complementation . The binary operations have been named and notated in various ways. Here they are called 'sum' and 'product', and notated by infix '+' and '∙', respectively. Sum and product commute and associate , as in

702-595: Is negation in both cases.) If 0=1, (1)-(3) are the axioms for an abelian group . (1) only serves to prove that concatenation commutes and associates. First assume that (1) associates from either the left or the right, then prove commutativity. Then prove association from the other direction. Associativity is simply association from the left and right combined. This basis makes for an easy approach to proof, called "calculation" in Laws of Form , that proceeds by simplifying expressions to 0 or 1, by invoking axioms (2)–(4), and

741-495: Is not a field, because ( 1 , 0 ) ⋅ ( 0 , 1 ) = ( 0 , 0 ) {\displaystyle (1,0)\cdot (0,1)=(0,0)} , but fields do not have zero divisors . Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism , namely any function compatible with

780-626: Is not defined for x = 0 ; or as an ordinary function whose value at 0 is arbitrary and must not be used. Simple structures : no binary operation : Group-like structures : one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. Ring-like structures or Ringoids : two binary operations, often called addition and multiplication , with multiplication distributing over addition. Lattice structures : two or more binary operations, including operations called meet and join , connected by

819-412: Is provided by the unary minus operation x ↦ − x . {\displaystyle x\mapsto -x.} Also, in universal algebra , a variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of

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858-547: Is read as True , '+' is read as OR , and '∙' as AND , and vice versa if 1 is read as False . These two operations define a commutative semiring , known as the Boolean semiring . 2 can be seen as grounded in the following trivial "Boolean" arithmetic: Note that: This Boolean arithmetic suffices to verify any equation of 2 , including the axioms, by examining every possible assignment of 0s and 1s to each variable (see decision procedure ). The following equations may now be verified: Each of '+' and '∙' distributes over

897-508: The absorption law . Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology . The added structure must be compatible, in some sense, with the algebraic structure. Algebraic structures are defined through different configurations of axioms . Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by identities and structures that are not. If all axioms defining

936-410: The allowed operations. The study of varieties is an important part of universal algebra . An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra ") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra T . Given

975-682: The associative law, but fail to satisfy the commutative law. Sets with one or more operations that obey specific laws are called algebraic structures . When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher arity operations) and operations that take only one argument ( unary operations ) or even zero arguments ( nullary operations ). The examples listed below are by no means

1014-417: The collection of all structures of a given type and homomorphisms between them form a concrete category . Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, a + ( b + c ) = ( a + b ) + c and a ( bc ) = ( ab ) c are associative laws , and

1053-603: The distinction [REDACTED] Index of articles associated with the same name This set index article includes a list of related items that share the same name (or similar names). If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Boolean-valued&oldid=1220163022 " Categories : Set index articles on mathematics Boolean algebra Logic and statistics Hidden categories: Articles with short description Short description

1092-462: The early years of the computer era. Perhaps the best of the lot, and one still in print, is: The following items reveal how the two-element Boolean algebra is mathematically nontrivial. Algebraic structure In mathematics , an algebraic structure consists of a nonempty set A (called the underlying set , carrier set or domain ), a collection of operations on A (typically binary operations such as addition and multiplication), and

1131-471: The elementary identities A A = A , A ¯ ¯ = A , 1 + A = 1 {\displaystyle AA=A,{\overline {\overline {A}}}=A,1+A=1} , and the distributive law. De Morgan's theorem states that if one does the following, in the given order, to any Boolean function : the result is logically equivalent to what you started with. Repeated application of De Morgan's theorem to parts of

1170-493: The equality must remain true. Here are some common examples. Some common axioms contain an existential clause . In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form "for all X there is y such that f ( X , y ) = g ( X , y ) {\displaystyle f(X,y)=g(X,y)} ", where X

1209-403: The free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings . Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields

Boolean-valued - Misplaced Pages Continue

1248-423: The operation(s) defining the structure. In this way, every algebraic structure gives rise to a category . For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms)

1287-486: The other: That '∙' distributes over '+' agrees with elementary algebra , but not '+' over '∙'. For this and other reasons, a sum of products (leading to a NAND synthesis) is more commonly employed than a product of sums (leading to a NOR synthesis). Each of '+' and '∙' can be defined in terms of the other and complementation: We only need one binary operation, and concatenation suffices to denote it. Hence concatenation and overbar suffice to notate 2 . This notation

1326-416: The sentence, "We have defined a ring structure on the set A {\displaystyle A} ", means that we have defined ring operations on the set A {\displaystyle A} . For another example, the group ( Z , + ) {\displaystyle (\mathbb {Z} ,+)} can be seen as a set Z {\displaystyle \mathbb {Z} } that

1365-453: The statement ( x = ∅) ∨ ( x = {0,1}) and is false when x is ⁠ { 1 } {\displaystyle \{1\}} ⁠ . The decidability for the first-order theory of many classes of Boolean algebras can still be shown, using quantifier elimination or small model property (with the domain size computed as a function of the formula and generally larger than 2). Many elementary texts on Boolean algebra were published in

1404-413: The statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of numbers , the additive inverse

1443-415: The usual algebra of real numbers . As for the order of operations , brackets are decisive if present. Otherwise '∙' precedes '+'. Hence A ∙ B + C is parsed as ( A ∙ B ) + C and not as A ∙ ( B + C) . Complementation is denoted by writing an overbar over its argument. The numerical analog of the complement of X is 1 − X . In the language of universal algebra , a Boolean algebra

1482-406: The validity of more general first-order logic formulas instead of only atomic positive equalities. As an example consider the formula ( x = 0) ∨ ( x = 1) . This formula is always true in a two-element Boolean algebra. In a four-element Boolean algebra whose domain is the powerset of ⁠ { 0 , 1 } {\displaystyle \{0,1\}} ⁠ , this formula corresponds to

1521-431: The vector space (called vectors ). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra . Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type ( homomorphisms ). In universal algebra, an algebraic structure

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