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Boolean-valued Function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = ), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate (logic)
In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula P(a), the symbol P is a predicate that applies to the individual constant a. Similarly, in the formula R(a,b), the symbol R is a predicate that applies to the individual constants a and b. According to Gottlob Frege, the meaning of a predicate is exactly a function from the domain of objects to the truth values "true" and "false". In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R(a,b) would be true on an interpretation if the entities denoted by a and b stand in the relation denoted by R. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation given to them. While first-order logic only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sentence (mathematical Logic)
In mathematical logic, a sentence (or closed formula)Edgar Morscher, "Logical Truth and Logical Form", ''Grazer Philosophische Studien'' 82(1), pp. 77–90. of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that ''must'' be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert R
Robert Lee Rayford (February 3, 1953 – May 15, 1969), sometimes identified as Robert R. due to his age, was an American teenager from Missouri who has been suggested to represent the earliest confirmed case of HIV/AIDS in North America. This is based on evidence published in 1988 in which the authors claimed that medical evidence indicated that he was "infected with a virus closely related or identical to human immunodeficiency virus type 1." Rayford died of pneumonia, but his other symptoms baffled the doctors who treated him. A study published in 1988 reported the detection of antibodies against HIV. Results of testing for HIV genetic material were reported at a scientific conference in Australia in 1999. However, the data has never been published in a peer-reviewed medical or scientific journal. No photos of Rayford are known to exist. Background Robert Rayford was born on February 3, 1953, in St. Louis, Missouri. As a single parent, his mother Constance had to rais ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zvi Kohavi
Zvi ( and , ''Tzvi'', Ṣvi, "gazelle") is a Jewish masculine given name. It is sometimes paired with Hirsch, the German and Yiddish word for "deer", in a bilingual pleonasm. Notable people with this name include: * Zvi Aharoni (1921–2012), Israeli Mossad agent * Zvi Arad (1942–2018), Israeli mathematician, acting president of Bar-Ilan University, president of Netanya Academic College * Zvi Hirsch Gregor Belkovsky (1865–1948), jurist, economist, and Zionist activist * Zvi Ben-Avraham (born 1941), Israeli geophysicist * Zvi Bern (born 1960), American physicist * Zvi Bodie (born 1943), American academic * Zvi Bornstein (1926–2024), Slovak antifascist fighter * Zvi Hirsch Chajes (1805–1855), Orthodox Polish rabbi * Zvi Chalamish, Israeli financier * Zvi Elpeleg (1926–2015), Israeli academic * Zvi Galil (born 1947), Israeli computer scientist, mathematician, and President of Tel Aviv University * Zvika Greengold (born 1952), Israeli officer during the Yom Kippu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Markham Brown
Frank, FRANK, or Franks may refer to: People * Frank (given name) * Frank (surname) * Franks (surname) * Franks, a Germanic people in late Roman times * Franks, a term in the Muslim world for all western Europeans, particularly during the Crusades Currency * Liechtenstein franc or frank, the currency of Liechtenstein since 1920 * Swiss franc or frank, the currency of Switzerland since 1850 * Westphalian frank, currency of the Kingdom of Westphalia between 1808 and 1813 * The currencies of the German-speaking cantons of Switzerland (1803–1814): ** Appenzell frank ** Aargau frank ** Basel frank ** Berne frank ** Fribourg frank ** Glarus frank ** Graubünden frank ** Luzern frank ** Schaffhausen frank ** Schwyz frank ** Solothurn frank ** St. Gallen frank ** Thurgau frank ** Unterwalden frank ** Uri frank ** Zürich frank Places * Frank, Alberta, Canada, an urban community, formerly a village * Franks, Illinois, United States, an unincorporated community * Franks ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory. A Boolean function takes the form f:\^k \to \, where \ is known as the Boolean domain and k is a non-negative integer called the arity of the function. In the case where k=0, the function is a constant element of \. A Boolean function with multiple outputs, f:\^k \to \^m with m>1 is a vectorial or ''vector-valued'' Boolean function (an S-box in symmetric cryptography). There are 2^ different Boolean functions with k arguments; equal to the number of different truth tables with 2^k entries. Every k-ary Boolean function can be expressed as a propositional formula in k variables x_1,...,x_k, and two propositional formulas a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proposition
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky is blue" expresses the proposition that the sky is blue. Unlike sentences, propositions are not linguistic expressions, so the English sentence "Snow is white" and the German "Schnee ist weiß" denote the same proposition. Propositions also serve as the objects of belief and other propositional attitudes, such as when someone believes that the sky is blue. Formally, propositions are often modeled as functions which map a possible world to a truth value. For instance, the proposition that the sky is blue can be modeled as a function which would return the truth value T if given the actual world as input, but would return F if given some alternate world where the sky is green. However, a number of alternative formalizations have be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Minimization
Logic optimization is a process of finding an equivalent representation of the specified logic circuit under one or more specified constraints. This process is a part of a logic synthesis applied in digital electronics and integrated circuit design. Generally, the circuit is constrained to a minimum chip area meeting a predefined response delay. The goal of logic optimization of a given circuit is to obtain the smallest logic circuit that evaluates to the same values as the original one. Usually, the smaller circuit with the same function is cheaper, takes less space, consumes less power, has shorter latency, and minimizes risks of unexpected cross-talk, hazard of delayed signal processing, and other issues present at the nano-scale level of metallic structures on an integrated circuit. In terms of Boolean algebra, the optimization of a complex Boolean expression is a process of finding a simpler one, which would upon evaluation ultimately produce the same results as the or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, A XOR B). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values. A proposition's truth table is a graphical representation of its truth function. The truth function can be more useful for mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be Truth value, true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of Logical conjunction, conjunction, Logical disjunction, disjunction, Material conditional, implication, Logical biconditional, biconditional, and negation. Some sources include other connectives, as in the table below. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or Quantifier (logic), quantifiers. However, all the machinery of pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Logic
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as Logical conjunction, conjunction (''and'') denoted as , disjunction (''or'') denoted as , and negation (''not'') denoted as . Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his ''An Investigation of the Laws of Thought'' (1854). According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
