Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 30
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. , fn ), with each fi a morphism of Ci ; morphism composition is definedcomponentwise: (f1 , . . . , fn ) ◦ (g1 , . . . , gn ) =(f1 ◦ g1 , . . . , fn ◦ gn ). The product of an arbitrary number of categories is defined similarly.product metric The metric on a finite product of metric spaces, defined by the formulad((x1 , . . . , xn ), (y1 , . . .
, yn ))= ni=1 di (x2ii,yi ) ,where di is a metric on Xi and (x1 , . . . , xn ),(y1 , . . . , yn ) ∈ X1 ×· · ·×Xn . See metric space.This definition shows that a finite product ofmetrizable spaces is metrizable.product of cardinal numbersThe productof some cardinal numbers is the cardinal numberthat is equinumerous with the Cartesian productof the given cardinal numbers. If κ and λ arecardinal numbers, k · λ denotes their cardinalproduct. For example, 3 · 2 = 6 and ℵ3 · ℵ17 =ℵ17 .product of objectsSuppose C is a categoryand {Ai : i ∈ I } is a family of objects of C,where I is some index set.
Let pi : A → Ai bea morphism for each i ∈ I . The tuple (A; pi :i ∈ I ) is the product of this family if, for every object B of C and every set of morphismsfi : B → Ai , for i ∈ I , there is a unique morphism f : B → A such that pi ◦ f = fi , forall i ∈ I . The morphisms pi are usually calledprojection morphisms.product spaceThe Cartesian product ofan arbitrary collection of topological spaces{Xα }α∈A , with the product topology. See product topology.product topologyThe standard topology forthe product α Xα of topological spaces. A basis is given by sets of the form α Uα where Uαis an open subset of Xα and Uα = Xα for allbut finitely many indices α.
The product topology is the coarsest topology on the product spacewhich makes all the projection maps continuous.See projection map.projection map The maps pα from the Cartesian product α∈A Xα of topological spaces,© 2001 by CRC Press LLCinto Xα , defined, for each α, by pα ({xα }) = xα .See also product space, product topology.projective geometryAn axiomatic systemthat grew out of perspective drawing during theRenaissance; one characterization axiom saysthat any two lines in the projective plane haveexactly one point in common. Analytically, theprojective space Pn of dimension n can be givenprojective coordinates (x0 : .
. . : xn ) which determine a point up to rescaling (λx0 : . . . : λxn )by a non-zero number λ, and satisfy the condition that at least one xi is non-zero. Thus, theset of lines containing the origin in R3 gives amodel for the projective plane P2 . In it, linescorrespond to planes through the origin and thepoint of intersection to the line common to twoplanes.projective set The projective sets form a hierarchy extending the Borel hierarchy in any Polish space X. Let 11 denote the collection of allanalytic sets in X. For n ≥ 1, let1n = {A ⊆ X : X \ A ∈ n1 }1and then let n+1be the collection of all pro1jections of n sets in X × NN , where NN is theBaire space. Let 1n = n1 ∩ 1n . Then the setsinP=n1 =1nnnare the projective sets.
They form a hierarchybecause for each n ≥ 1,1n1 ∪ 1n ⊆ 1n+1 = n+1∩ 1n+1 .In addition,11 = Borel .The projective classes n1 , 1n , and 1n are alsoknown as Lusin point classes.proofIn first order logic, let L be a first order language and consider a particular predicatecalculus for L, with the set of logical axioms. Let α be a well-formed formula of L.A proof of α in the predicate calculus is a sequence α1 , α2 , . .
. , αn of well-formed formulasof L such that αn = α and such that for all i,1 ≤ i ≤ n, either(i.) αi ∈ (i.e., αi is a logical axiom) orpropositional logic(ii.) there exist 1 ≤ j1 < · · · < jk < i suchthat αi can be deduced from αj1 , . . . , αjk usinga rule of inference. (The value of k depends onthe rule of inference; for example, if the rule ofinference is modus ponens, then k = 2.)The formula α is provable from the predicatecalculus (notation: α) if there is a proof of αfrom the predicate calculus.If is a set of well-formed formulas of L,then a proof of α from is a sequence α1 , α2 ,.
. . , αn of well-formed formulas of L such thatαn = α and such that for all i, 1 ≤ i ≤ n, either(i.) αi ∈ ∪ or(ii.) there exist 1 ≤ j1 < · · · < jk < i suchthat αi can be deduced from αj1 , . . . , αjk usinga rule of inference.The formula α is provable from in the predicate calculus (notation: α) if there is aproof of α from in the predicate calculus.The notion of proof in propositional logic isentirely analogous.The notion of proof in formal logic is alsocalled formal proof or deduction.proper fractionA positive rational numberawhereaandbarepositiveintegers and a < b.b15For example, 38 is a proper fraction, while 158is an improper fraction.properly discontinuous transformation groupA group G acts properly discontinuously on aspace X if, for each x in X, there is an openneighborhood U so that whenever g is not theidentity in G, U ∩ gU is empty.Properly discontinuous transformation groupsare useful for studying covering spaces, whichare given by maps X −→ Y such that any pointin Y has a (connected) neighborhood U whoseinverse image is the disjoint union of open setsof X each homeomorphic to U .
A particular instance of this is the covering of a homogeneousspace by a Lie group: an inclusion of Lie groupsH −→ G has coset space G/H which inheritsthe quotient topology from G; the quotient mapG −→ G/H is a covering map.proper subset (of a set)A set S is a propersubset of a set T if S is a subset of T but S is notequal to T .
Thus, a set is never a proper subset ofitself. For example, {3, 10} is a proper subset of{3, 10, 47}. The notation is not entirely uniform.© 2001 by CRC Press LLCIf S is a proper subset of T , many authors denotethis by S ⊂ T , whereas others choose to denote⊂⊂T or S−T , for example. Compare withit by S=subset.propositional calculusThe syntactical partof propositional logic. A propositional calculusis a formal system, consisting of an alphabet (seepropositional logic), the set of all well-formedformulas, a particular set of well-formed formulas, which are called axioms, and a list of rulesof deduction.The well-formed formulas that are axiomsare formulas that are intuitively obvious, andshould be tautologies.
A typical axiom thatmight occur in a propositional calculus wouldbe(α → (β → α)),where α and β are any well-formed propositional formulas (this is actually called an axiomscheme, rather than an axiom, since there are infinitely many axioms of this form, one for eachdifferent choice of α and β). A typical rule ofdeduction in a propositional calculus is modusponens.
See modus ponens.A propositional calculus, using the axiomsand rules of deduction, is used to prove theorems. See also proof, theorem. While the actualchoice of axioms and rules of deduction is notimportant, it is important that a propositionalcalculus be both sound (i.e., any well-formedformula that can be proved from the formal system should be a tautology) and complete (i.e.,any tautology should be provable from the formal system).Propositional calculus is also called sentential, or statement calculus.propositional logicA formal logic with thefollowing alphabet of symbols:(i.) (, ) (parentheses)(ii.) ¬, ∨, ∧, →, ↔ (logical connectives)(iii.) A1 , A2 , A3 , .
. . (nonlogical symbols).Often the list of logical connectives in item(ii.) is shortened to some complete list of logicalconnectives, such as {¬, →}. The symbol Anis called the nth propositional symbol, or nthsentential or sentence symbol.The propositional symbols have no meaning, although using truth assignments they canpseudocompact topological spacebe interpreted as either true or false. Propositional logic has rules that tell which expressions from the language are well-formed formulas. The propositional calculus (sententialcalculus) is used to produce theorems of propositional logic. See propositional calculus, proof.Truth assignments lead to a semantic notion oftruth in propositional logic, while the propositional calculus gives a syntactical notion of provability. Propositional logic is also called sentential logic.pseudocompact topological spaceA topological space X with the property that everyreal-valued continuous function defined on Xis bounded.
Pseudocompact spaces play a significant role in the theory of C ∗ -algebras.pseudomanifoldA simplicial complex Swhich is a union of n-simplices (for some n)and satisfies(i.) each n − 1-simplex of S is the face ofexactly two n-simplices, and(ii.) given two n-simplices σ and σ thereexists a sequence of n-simplices σ = σ0 , σ1 ,.
. . , σk = σ with σi ∩ σi+1 an n − 1 simplexof S.Psuedomanifolds share a key homological property with actual manifolds; namely, Hn (S) = Zif S is orientable and Hn (S) = 0 otherwise.pseudoprime (1) An odd composite integer,n, with the property that 2n ≡ 2 (mod n). Thatis, n is a pseudoprime if n is a divisor of 2n − 2.© 2001 by CRC Press LLCThe name is derived from the fact that if p isa prime number, then a p ≡ a (mod p) for allintegers a.(2) A composite integer n so that a n ≡ a(mod n) for all integers a is an absolute pseudoprime.Pythagorean fieldA field, F , in which thesum of the squares of any two elements from thefield is the square of an element from the field.That is, F is Pythagorean if, for every a and b inF , there exists a c in F so that a 2 +b2 = c2 .
Therational numbers are not Pythagorean since 12 +12 = 2 is not the square of a rational number.However, the√ real numbers are a Pythagoreanfield since a 2 + b2 is a real number whenevera and b are real.Pythagorean triple A triple of positive integers (a, b, c) satisfying the equation a 2 + b2 =c2 . For example, (6, 8, 10) is a Pythagoreantriple since 62 + 82 = 102 . If (a, b, c) is aPythagorean triple and a, b, and c are pairwiserelatively prime, then (a, b, c) is known as aprimitive Pythagorean triple ((3, 4, 5) and(5, 12, 13) are primitive Pythagorean triples).It can be shown that either a or b (or both)must be even if (a, b, c) is a Pythagorean triple.In fact, (a, b, c) is a Pythagorean triple if andonly if there exist positive integers k, m, and nso that gcd(m, n) = 1, exactly one of m or nis even, a = (m2 − n2 )k, b = 2mnk, and c =(m2 + n2 )k, providing a formula for generatingall Pythagorean triples.quotient spaceQquantifierQuantifiers are used in order toquantify if elements with a certain property existin a particular universe.
The quantifiers are denoted symbolically by ∃ (the existential quantifier) and ∀ (the universal quantifier). The interpretation of the existential quantifier (∃x)[. . . ] isthat there exists an object x (possibly more thanone) in the universe with property [. . . ]. The interpretation of the universal quantifier (∀x)[. . . ]is that all objects x in the universe have property[. .
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