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Equality (mathematics)
In mathematics, equality is a relationship between two quantities or Expression (mathematics), expressions, stating that they have the same value, or represent the same mathematical object. Equality between and is written , and read " equals ". In this equality, and are distinguished by calling them ''sides of an equation, left-hand side'' (''LHS''), and ''right-hand side'' (''RHS''). Two objects that are not equal are said to be distinct. Equality is often considered a primitive notion, meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothing else"), reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality like Reflexive relation, reflexivity, Symmetric relation, symmetry, and Transitive relation, transitivity have been understood intuitively since at least the ancient Greeks, but were not symboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equals Sign Typewriter
Equal(s) may refer to: Mathematics * Equality (mathematics). * Equals sign (=), a mathematical symbol used to indicate equality. Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a board game * The Equals, a British pop group formed in 1965 * "Equal", a 2016 song by Chrisette Michele from ''Milestone (Chrisette Michele album), Milestone'' * "Equal", a 2022 song by Odesza featuring Låpsley from ''The Last Goodbye (album), The Last Goodbye'' * "Equals", a 2009 song by Set Your Goals from ''This Will Be the Death of Us'' * Equal (TV series), ''Equal'' (TV series), a 2020 American docuseries on HBO * = (album), ''='' (album), a 2021 album by Ed Sheeran * "=", a 2022 song by J-Hope from ''Jack in the Box (album), Jack in the Box'' Other uses * Equal (sweetener), a brand of artificial sweetener. * EQUAL Community Initiative, an initiative within the European Social Fund of the European Union. See also [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recorde - The Whetstone Of Witte - Equals
Robert Recorde () was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus (+) and minus (−) signs to English speakers in 1557. Biography Born around 1510, Robert Recorde was the second and last son of Thomas and Rose Recorde of Tenby, Pembrokeshire, in Wales. Recorde entered the University of Oxford about 1525, and was elected a Fellow of All Souls College there in 1531. Having adopted medicine as a profession, he went to the University of Cambridge to take the degree of M.D. in 1545. He afterwards returned to Oxford, where he publicly taught mathematics, as he had done prior to going to Cambridge. He invented the "equals" sign, which consists of two horizontal parallel lines, stating that no two things can be more equal. It appears that he afterwards went to London, and acted as physician to King Edward VI and to Queen Mary, to whom some of his books are dedicated. He was also controller of the Royal Mint and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Equation Ever
First most commonly refers to: * First, the ordinal form of the number 1 First or 1st may also refer to: Acronyms * Faint Images of the Radio Sky at Twenty-Centimeters, an astronomical survey carried out by the Very Large Array * Far Infrared and Sub-millimetre Telescope, of the Herschel Space Observatory * For Inspiration and Recognition of Science and Technology, an international youth organization * Forum of Incident Response and Security Teams, a global forum Arts and entertainment Albums * ''1st'' (album), by Streets, 1983 * ''1ST'' (SixTones album), 2021 * ''First'' (David Gates album), 1973 * ''First'', by Denise Ho, 2001 * ''First'' (O'Bryan album), 2007 * ''First'' (Raymond Lam album), 2011 Extended plays * ''1st'', by The Rasmus, 1995 * ''First'' (Baroness EP), 2004 * ''First'' (Ferlyn G EP), 2015 Songs * "First" (Lindsay Lohan song), 2005 * "First" (Cold War Kids song), 2014 * "First", by Lauren Daigle from the album '' How Can It Be'', 2015 * "First" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Extensionality
The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the two set (mathematics), sets ''A'' and ''B'' are equal if and only if ''A'' and ''B'' have the same members. Etymology The term ''extensionality'', as used in '''Axiom of Extensionality has its roots in logic. An intensional definition describes the necessary and sufficient conditions for a term to apply to an object. For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -6, -8..." In logic, the Extension (logic), extension of a Predicate (mathematical logic), predicate is the set of all things for which the predicate is true. The logical term was introduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Element (mathematics)
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called containing the first four positive integers (A = \), one could say that "3 is an element of ", expressed notationally as 3 \in A . Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets of . Sets can themselves be elements. For example, consider the set B = \. The elements of are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set \. The elements of a set can be anything. For example the elements of the set C = \ are the color red, the number 12, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundation Of Mathematics
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality. The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's ''Elements''. A mathematical assertion is considered as truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms ( inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulates. These foundations were tacitly assumed to be definitive until the introduction of infinitesimal calculus by Isaac New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundational Crisis In Mathematics
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality. The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's ''Elements''. A mathematical assertion is considered as truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms ( inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulates. These foundations were tacitly assumed to be definitive until the introduction of infinitesimal calculus by Isaac Newto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Identity
In logic, the law of identity states that each thing is identical with itself. It is the first of the traditional three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws. History Ancient philosophy The earliest recorded use of the law appears in Plato's dialogue '' Theaetetus'' (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing: It is used explicitly only once in Aristotle, in a proof in the '' Prior Analytics'': Medieval philosophy Aristotle believed the law of non-contradiction to be the most fundamental law. Both Thomas Aquinas (''Met.'' IV, lect. 6) and Duns Scotus (''Quaest. sup. Met.'' IV, Q. 3) follow Aristotle in this respect. Antonius Andreas, the Spanish disciple of Scotus (d. 1320), argues that the first place should belong to the law "Every Being is a Being" (''Omne Ens est Ens'', Qq. in Met ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. A ''free variable'' is a Mathematical notation, notation (symbol) that specifies places in an expression (mathematics), expression where Substitution (logic), substitution may take place and is not a parameter of this or any container expression. The idea is related to a ''placeholder'' (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variable (programming), variables used in a function (computer science), function that are neither local variables nor parameter (computer programming), parameters of that function. The term non-local variable is often a synonym in this co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
