"algorithm and Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes' theorem ""is to the theory of probability what the Pythagorean theorem is to geometry"". Bayes' theorem is stated mathematically as the following equation: where formula_2 and formula_3 are events and formula_4. Suppose that a test for using a particular drug is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a"
"Theorem In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally ""deductive"", in contrast to"
"Existence theorem In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all , , ... there exist(s) ...'. That is, in more formal terms of symbolic logic, it is a theorem with a prenex normal form involving the existential quantifier. Many such theorems will not do so explicitly, as usually stated in standard mathematical language – for example, the statement that the sine function is continuous; or any theorem written in big O notation. The quantification can be found in the definitions of the concepts used. A controversy that goes"
"A theorem is a statement that can be derived from those axioms by application of these rules of inference. Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood). Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge"
"example in which the hypothesis is """"n"" is an even natural number"" and the conclusion is """"n""/2 is also a natural number"". To be proved, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of"
"between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically, many theorems are of the form of an indicative conditional: ""if A, then B"". Such a theorem does not assert ""B"", only that ""B"" is a necessary consequence of ""A"". In this case ""A"" is called the hypothesis of the theorem (""hypothesis"" here is something very different from a conjecture) and ""B"" the conclusion (formally, ""A"" and ""B"" are termed the ""antecedent"" and ""consequent""). The theorem ""If ""n"" is an even natural number then ""n""/2 is a natural number"" is a typical"
"from a ""term"" (which we can see as the computational content). Existence theorem In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all , , ... there exist(s) ...'. That is, in more formal terms of symbolic logic, it is a theorem with a prenex normal form involving the existential quantifier. Many such theorems will not do so explicitly, as usually stated in standard mathematical language – for example, the statement that the sine function is continuous; or any theorem written in big O notation. The quantification can be found"
"Lagrange's theorem (number theory) In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if ""p"" is a prime number and formula_1 is a polynomial with integer coefficients, then either: Solutions are ""incongruent"" if they do not differ by a multiple of ""p"". If the modulus is not prime, then it is possible for there to be more than deg ""f(x)"" solutions. The two key ideas are the following. Let formula_3 be the polynomial obtained from"
"theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval. More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of ƒ exists for almost every ""x"" and that the (almost everywhere defined) limit function ƒ̂ is integrable: Furthermore, formula_4 is ""T""-invariant, that is to say holds almost everywhere, and if ""μ""(""X"") is finite, then the normalization is the same: In particular, if ""T"" is ergodic, then ƒ̂ must be a constant (almost everywhere), and so one has that almost everywhere."
"Theory (mathematical logic) In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element formula_1 of a theory formula_2 is then called an axiom of the theory, and any sentence that follows from the axioms (formula_3) is called a theorem of the theory. Every axiom is also a theorem. A first-order theory is a set of first-order sentences. When defining theories for foundational purposes, additional care must be taken and normal set-theoretic language may not be appropriate. The construction of a theory"
"Uniqueness theorem In mathematics, a uniqueness theorem is a theorem proving that certain conditions determine a unique solution. Examples of uniqueness theorems include: A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress"
"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. For example, The theorem says two things for this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two"
"by statements like ""An [""object""] is [""word to be defined""] if [""condition""]"" (for example, ""A set is closed if it contains all of its limit points.""). As a special convention, the word ""if"" in such a definition should be interpreted as ""if and only if"". Theorems have generally a title or label in bold type, and possibly identify the originator (for example, """"). This is immediately followed by the statement of the theorem, usually set in italics. The proof of a theorem is usually clearly delimited, starting with the word ""Proof"" while the end of the proof is indicated by"
"For example, is a derivation. Therefore, ""ABBBAB"" is a theorem of formula_8 The notion of truth (or falsity) cannot be applied to the formula ""ABBBAB"" until an interpretation is given to its symbols. Thus in this example, the formula does not yet represent a proposition, but is merely an empty abstraction. Two metatheorems of formula_3 are: Theorem In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical"
"under the theory. Sometimes two theories have exactly the same explanatory power because they make the same predictions. A pair of such theories is called indistinguishable or observationally equivalent, and the choice between them reduces to convenience or philosophical preference. The form of theories is studied formally in mathematical logic, especially in model theory. When theories are studied in mathematics, they are usually expressed in some formal language and their statements are closed under application of certain procedures called rules of inference. A special case of this, an axiomatic theory, consists of axioms (or axiom schemata) and rules of inference."
"representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory. A theorem may be expressed in a formal language (or ""formalized""). A formal theorem is the purely formal analogue of a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntactic conditions. The notation formula_1 is often used to indicate that formula_1 is a theorem. Formal theorems consist of formulas of a formal language and the transformation rules of a formal system. Specifically, a formal theorem is always the"
"argument being a term. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model. The well-formed terms and propositions of ordinary first-order logic have the following syntax: Terms: that is, a term is recursively defined to be a constant ""c"" (a named object from the domain of discourse), or a variable ""x"" (ranging over the objects in the domain of discourse), or an ""n""-ary function ""f"" whose arguments are terms ""t"". Functions map tuples of objects to objects. Propositions: that is, a proposition"
"theories include Peano arithmetic and the weaker Robinson arithmetic. A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all computable functions. Let ""T"" be a first-order theory in the language of arithmetic and capable of representing all computable functions. Let F be a formula in the language with one free variable, then: Intuitively, ψ is a self-referential sentence saying that ψ has the property F. The sentence ψ can also be viewed as a fixed point of the operation assigning to each formula θ the sentence F(°#(θ)). The sentence ψ constructed"
"required to establish the status of a statement as a theorem. The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs. Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems,"
"Brouwer. The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: This can be generalized to an arbitrary finite dimension: A slightly more general version is as follows: An even more general form is better known under a different name: The theorem holds only for sets that are ""compact"" (thus, in particular, bounded and closed) and ""convex"" (or homeomorphic to convex). The following examples show why the pre-conditions are important. Consider the function which is a continuous function from formula_9 to itself. As it"
"Addition theorem In mathematics, an addition theorem is a formula such as that for the exponential function that expresses, for a particular function ""f"", ""f""(""x"" + ""y"") in terms of ""f""(""x"") and ""f""(""y""). Slightly more generally, as is the case with the trigonometric functions ""sin"" and ""cos"", several functions may be involved; this is more apparent than real, in that case, since there ""cos"" is an algebraic function of ""sin"" (in other words, we usually take their functions both as defined on the unit circle). The scope of the idea of an addition theorem was fully explored in the nineteenth"
"Therefore, the definition of a theorem in system L is: An example of the proof of a sequent (Modus Tollendo Tollens in this case): An example of the proof of a sequent (a theorem in this case): Each rule of system L has its own requirements for the type of input(s) or entry(es) that it can accept and has its own way of treating and calculating the assumptions used by its inputs. The historical development of tabular-layout natural deduction systems, which are rule-based, and which indicate antecedent propositions by line numbers (and related methods such as vertical bars or asterisks)"
"the physical axioms on which such ""theorems"" are based are themselves falsifiable. A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary and the usage of some terms has evolved over time. There are other terms, less commonly used, that are conventionally attached to proved statements, so that certain theorems are referred to by historical or customary names. For example: A few well-known theorems have even more idiosyncratic names. The division algorithm (see Euclidean division) is a theorem expressing the outcome"
"interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a ""derivation"" or ""proof"" and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system,"
"roots of formula_13, which by (ii) is at most formula_29, which is at most formula_30. Lagrange's theorem (number theory) In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if ""p"" is a prime number and formula_1 is a polynomial with integer coefficients, then either: Solutions are ""incongruent"" if they do not differ by a multiple of ""p"". If the modulus is not prime, then it is possible for there to be more than deg ""f(x)"""
"rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus. The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then"
"equivalent to a relation algebra with two ordered pair projection functions. A first-order theory of a particular signature is a set of axioms, which are sentences consisting of symbols from that signature. The set of axioms is often finite or recursively enumerable, in which case the theory is called effective. Some authors require theories to also include all logical consequences of the axioms. The axioms are considered to hold within the theory and from them other sentences that hold within the theory can be derived. A first-order structure that satisfies all sentences in a given theory is said to be"
"were treated as theoretical concepts. In this view, theories function as axioms: predicted observations are derived from the theories much like theorems are derived in Euclidean geometry. However, the predictions are then tested against reality to verify the theories, and the ""axioms"" can be revised as a direct result. The phrase ""the received view of theories"" is used to describe this approach. Terms commonly associated with it are ""linguistic"" (because theories are components of a language) and ""syntactic"" (because a language has rules about how symbols can be strung together). Problems in defining this kind of language precisely, e.g., are"
"is the set of real numbers. The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms. The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields. Theory (mathematical logic) In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element formula_1 of a theory formula_2 is then called an axiom of the theory,"
"last formula of a derivation in some formal system each formula of which is a logical consequence of the formulas that came before it in the derivation. The initially accepted formulas in the derivation are called its axioms, and are the basis on which the theorem is derived. A set of theorems is called a theory. What makes formal theorems useful and of interest is that they can be interpreted as true propositions and their derivations may be interpreted as a proof of the truth of the resulting expression. A set of formal theorems may be referred to as a"
"description (i.e. a model) of external phenomena. A ""theorem"" is a symbol or string of symbols which is derived by using a formal system. The string of symbols is a logical consequence of the axioms and rules of the system. A ""formal proof"" or ""derivation"" is a finite sequence of propositions (called well-formed formulas in the case of a formal language) each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. The concept of natural deduction is"
"Factor theorem In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial formula_1 has a factor formula_2 if and only if formula_3 (i.e. formula_4 is a root). Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent. The factor theorem is also used to remove known zeros from a polynomial while"
"Deduction theorem In mathematical logic, the deduction theorem is a metatheorem of propositional and first-order logic. It is a formalization of the common proof technique in which an implication ""A"" → ""B"" is proved by assuming ""A"" and then deriving ""B"" from this assumption conjoined with known results. The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct. Though it has seemed ""obvious"" to mathematicians literally for centuries that proving ""B"" from ""A"" conjoined with a set of theorems is tantamount to proving the implication ""A"" → ""B"" based on those theorems alone, it was left"