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70 relations: Addison-Wesley, Addition, Al-Karaji, Arithmetic progression, Augustus De Morgan, Axiom, Axiom schema, Bhāskara II, Binomial theorem, Blaise Pascal, Cardinal number, Chakravala method, Charles Sanders Peirce, Combinatorial proof, Combinatorics, Computer science, Concrete Mathematics, Deductive reasoning, Euclid, Fibonacci number, First-order logic, Formal proof, Formal verification, Francesco Maurolico, Fundamental theorem of arithmetic, George Boole, Giuseppe Peano, Golden ratio, Harvard University, Inductive reasoning, Infinite descending chain, Ivor Grattan-Guinness, Jacob Bernoulli, Limit ordinal, Mathematical logic, Mathematical proof, Natural number, Ordinal number, Parmenides (dialogue), Pascal's triangle, Peano axioms, Philosophy, Pierre de Fermat, Plato, Polysyllogism, Prime number, Problem of induction, Proof by exhaustion, Proof by infinite descent, Proofs involving the addition of natural numbers, ..., Q.E.D., Range (mathematics), Recursion, Recursion (computer science), Reflexive relation, Richard Dedekind, Rule of inference, Second-order logic, Set theory, Simon Fraser University, Sorites paradox, Structural induction, Topology, Transfinite induction, Tree (set theory), Vacuous truth, Well-founded relation, Well-order, Well-ordering principle, Zermelo–Fraenkel set theory. Expand index (20 more) »
Addison-Wesley
Addison-Wesley is a publisher of textbooks and computer literature.
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Addition
Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.
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Al-Karaji
(c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad.
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Arithmetic progression
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
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Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician.
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
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Axiom schema
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
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Bhāskara II
Bhāskara (also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhaskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer.
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Binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.
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Blaise Pascal
Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, writer and Catholic theologian.
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Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
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Chakravala method
The chakravala method (चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation.
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Charles Sanders Peirce
Charles Sanders Peirce ("purse"; 10 September 1839 – 19 April 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism".
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Combinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof.
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
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Computer science
Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.
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Concrete Mathematics
Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatment of the analysis of algorithms.
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Deductive reasoning
Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.
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Euclid
Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
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Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.
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First-order logic
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
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Formal proof
A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference.
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Formal verification
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of mathematics.
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Francesco Maurolico
Francesco Maurolico (Greek: Φραγκίσκος Μαυρόλυκος, Frangiskos Mavrolikos; Latin: Franciscus Maurolycus; Francisci Maurolyci; Italian: Francesco Maurolico; 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily.
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Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
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George Boole
George Boole (2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland.
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Giuseppe Peano
Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.
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Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
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Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts.
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Inductive reasoning
Inductive reasoning (as opposed to ''deductive'' reasoning or ''abductive'' reasoning) is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion.
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Infinite descending chain
Given a set S with a partial order ≤, an infinite descending chain is an infinite, strictly decreasing sequence of elements x1 > x2 >...
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Ivor Grattan-Guinness
Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic.
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Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.
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Limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.
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Mathematical logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
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Mathematical proof
In mathematics, a proof is an inferential argument for a mathematical statement.
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
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Parmenides (dialogue)
Parmenides (Παρμενίδης) is one of the dialogues of Plato.
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Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.
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Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.
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Philosophy
Philosophy (from Greek φιλοσοφία, philosophia, literally "love of wisdom") is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.
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Pierre de Fermat
Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.
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Plato
Plato (Πλάτων Plátōn, in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world.
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Polysyllogism
A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, is a premise for the next, and so on.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Problem of induction
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense, highlighting the apparent lack of justification for.
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Proof by exhaustion
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds.
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Proof by infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction that relies on the least integer principle.
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Proofs involving the addition of natural numbers
This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity.
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Q.E.D.
Q.E.D. (also written QED and QED) is an initialism of the Latin phrase quod erat demonstrandum meaning "what was to be demonstrated" or "what was to be shown." Some may also use a less direct translation instead: "thus it has been demonstrated." Traditionally, the phrase is placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition has been restated exactly, as the conclusion of the demonstration or completion of the proof.
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Range (mathematics)
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.
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Recursion
Recursion occurs when a thing is defined in terms of itself or of its type.
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Recursion (computer science)
Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem (as opposed to iteration).
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Reflexive relation
In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.
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Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.
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Rule of inference
In logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions).
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Second-order logic
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Simon Fraser University
Simon Fraser University (SFU) is a public research university in British Columbia, Canada with campuses in Burnaby (Main Campus), Surrey, and Vancouver.
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Sorites paradox
The sorites paradox (sometimes known as the paradox of the heap) is a paradox that arises from vague predicates.
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Structural induction
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
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Tree (set theory)
In set theory, a tree is a partially ordered set (T, \omega + 1.
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Vacuous truth
In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.
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Well-founded relation
In mathematics, a binary relation, R, is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.
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Well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
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Well-ordering principle
In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element.
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Zermelo–Fraenkel set theory
In mathematics, 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.
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References
[1] https://en.wikipedia.org/wiki/Mathematical_induction
