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Falsum
"Up tack" is the Unicode name for a symbol (⊥, \bot in LaTeX, U+22A5 in Unicode) that is also called "bottom", "falsum", "absurdum", or "the absurdity symbol", depending on context. It is used to represent: * The truth value false (logic), 'false', or a logical constant denoting a proposition in logic that is always false. (The names "falsum", "absurdum" and "absurdity symbol" come from this context.) * The bottom element in wheel theory and lattice theory, which also represents absurdum when used for logical semantics * The bottom type in type theory, which is the bottom element in the subtype relation. This may coincide with the empty type, which represents absurdum under the Curry–Howard correspondence * The "undefined value" in quantum physics interpretations that reject counterfactual definiteness, as in (r0,⊥) as well as * Mixed radix decoding in the APL (programming language), APL programming language The glyph of the up tack appears as an upside-down Tee (symbol), te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contradiction (logic)
In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." In modern formal logic and type theory, the term is mainly used instead for a ''single'' proposition, often denoted by the falsum symbol Bottom type, \bot; a proposition is a contradiction if false (logic), false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). This can be generalized to a collection of propositions, which is then said to "contain" a contradiction. History By creation of a paradox, Plato's ''Euthydemus (dialogue), Euthydemus'' dialogue demonstrates the need for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
False (logic)
In logic, false (Its noun form is falsity) or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, O''pq''), and the up tack symbol \bot. Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), \bot, is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity. In classical logic and Boolean logic In Boolean logic, each variable denotes a truth value which can be either true (1), or false (0). In a classical propositional calculus, each proposition will be assigned a truth value of either ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in computing as well as various types of logic. Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called falsy and truthy. For example, in Lisp, nil, the empty list, is treated as false, and all other values are treated as true. In C, the number 0 or 0.0 is false, and all other values are treated as true. In JavaScript, the empty string (""), null, undefined, NaN, +0, −0 and false are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. ''Perpendicular'' is also used as a noun: a perpendicular is a line which is perpendicular to a given line or plane. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plus And Minus Signs
The plus sign () and the minus sign () are Glossary of mathematical symbols, mathematical symbols used to denote sign (mathematics), positive and sign (mathematics), negative functions, respectively. In addition, the symbol represents the operation of addition, which results in a Sum (mathematics), sum, while the symbol represents subtraction, resulting in a difference (mathematics), difference. Their use has been extended to many other meanings, more or less analogous. and are Latin terms meaning 'more' and 'less', respectively. The forms and are used in many countries around the world. Other designs include for plus and for minus. History Though the signs now seem as familiar as the alphabet or the Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembles a pair of legs walking in the direction in which the text was written (Egyptian language, Egyptian could be written either from right to left or left to r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Independence
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If A is the hypothesis, and B and C are observations, conditional independence can be stated as an equality: :P(A\mid B,C) = P(A \mid C) where P(A \mid B, C) is the probability of A given both B and C. Since the probability of A given C is the same as the probability of A given both B and C, this equality expresses that B contributes nothing to the certainty of A. In this case, A and B are said to be conditionally independent given C, written symbolically as: (A \perp\!\!\!\perp B \mid C). The concept of conditional independence is essential to graph-based theories of statistical inference, as it estab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Operators And Symbols In Unicode
The Unicode Standard encodes almost all standard characters used in mathematics. Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation. Mathematical operators and symbols are in multiple Unicode blocks. Some of these blocks are dedicated to, or primarily contain, mathematical characters while others are a mix of mathematical and non-mathematical characters. This article covers all Unicode characters with a derived property of "Math". Dedicated blocks Mathematical Operators block The Mathematical Operators block (U+2200–U+22FF) contains characters for mathematical, logical, and set notation. Supplemental Mathematical Operators block The Supplemental Mathematical Operators block (U+2A00–U+2AFF) contains various mathematical symbols, including N-ary operators, summations and integrals, intersections and unions, logical and relational operators, and subset/superset relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of XML And HTML Character Entity References
In SGML, HTML and XML documents, the logical constructs known as ''character data'' and ''attribute values'' consist of sequences of characters, in which each character can manifest directly (representing itself), or can be represented by a series of characters called a ''character reference'', of which there are two types: a ''numeric character reference'' and a ''character entity reference''. This article lists the character entity references that are valid in HTML and XML documents. A character entity reference refers to the content of a named entity. An entity declaration is created in XML, SGML and HTML documents (before HTML5) by using the syntax in a document type definition (DTD). Character reference overview In HTML and XML, a ''numeric character reference'' refers to a character by its Universal Coded Character Set/Unicode ''code point'', and uses the format: &#x''hhhh''; or &#''nnnn''; where the x must be lowercase in XML documents, ''hhhh'' is the code po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprimality
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variables
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which * the domain is the set of possible outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
