Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 40

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 40)

There is ananalogous sequence for homology. One can usethe Wang sequence to compute the homology ofthe based loop space of a sphere.wedge The one-point union of two spaces; inother words, the wedge product of two spaces isformed from their disjoint union by identifyingone chosen point in the first space with a chosenpoint in the second. In the category of pointedspaces (spaces together with a base point), thechosen point is the base point. For example, thewedge of two circles is a figure eight.well-formed formula In propositional (sentential) logic, a well-formed formula (or wff) satisfies the following inductive definition.(i.) If A is a sentence symbol, then A is a wff.(ii.) If α and β are wffs, then so are (¬α),(α ∧ β), (α ∨ β), (α → β), and (α ↔ β).(iii.) The set of well-formed formulas is generated by rules (i.) and (ii.).For example, if A, B, and C are sentencesymbols, then ((A ∧ B) ∨ C) is a wff, whileA∧ is not a wff.

Informally, the parenthesesused in defining wffs are often omitted whendoing so does not affect the readability of theformula; in particular, it is always assumed that¬, ∧, and ∨ apply to as little as possible. Forexample, if A, B, and C are sentence symbols,then ¬A ∧ B → C means (((¬A) ∧ B) → C).In first order logic, with a given first orderlanguage L, the set of wffs of L is defined inductively.© 2001 by CRC Press LLC(i.) If α is an atomic formula, then α is a wff.(ii.) If α and β are wffs, then so are (¬α) and(α → β).(iii.) If α is a wff and v is a variable, then∀vα is a wff.(iv.) The set of well-formed formulas is generated by rules (i.), (ii.), and (iii.).Since {¬, →} is a complete set of logicalconnectives, it is possible to use the other connectives informally in well-formed formulas asabbreviations for formulas in the actual formallanguage L.

In particular, if α and β are wellformed formulas of L, then(i.) (α ∨ β) abbreviates ((¬α) → β).(ii.) (α ∧ β) abbreviates (¬(α → (¬β))).(iii.) (α ↔ β) abbreviates ((α → β)∧(β →α)).Informally, the parentheses used in definingwffs are often omitted when doing so does not affect the readability of the formula, or even addedwhen doing so aids the readability of the formula. It is always assumed that ∀ applies to aslittle as possible. For example, ∀vα → β means(∀vα → β), rather than ∀v(α → β).For example, in the language of elementarynumber theory (see first order language), ∀v1 (<(v1 , S(v1 ))) is a well-formed formula, although< (v1 , S(v1 )) is usually informally written asv1 < S(v1 ).well-founded relation A partial ordering R,on a set S, such that every nonempty subset ofS has an R-minimal element.

For example, therelation “m divides n”, on the set of natural numbers, is well-founded; the relation ≤ on the setof real numbers is not well founded.well-founded set A set X on which the membership relation is well founded. That is, anynonempty subset of X contains an -minimalelement. A well-founded set cannot contain itself as a member.well-ordered setA pair (S, ≤) such that ≤is a well-ordering of S. For example, (N, ≤) isa well-ordered set. Also called woset.well-orderingA linear ordering ≤ of someset S such that every nonempty subset of S hasa minimum element. For example, the usualWell-Ordering Theoremlinear ordering ≤ for numbers is a well-orderingof N but it is not a well-ordering of R.Well-Ordering TheoremEvery set can bewell ordered; i.e., for every set there exists an ordering on that set which is a well-ordering. Seewell-ordering.

The Well-Ordering Theorem isequivalent to the Axiom of Choice. See Axiomof Choice. Consequently, the Well-OrderingTheorem is independent of the axioms of ZF(Zermelo-Fraenkel set theory); that is, it can neither be proved nor disproved from ZF .© 2001 by CRC Press LLCWhitney sumThe sum of two vector bundles over a manifold, formed by taking the directsum of the vector spaces over each point.

TheMöbius band M can be thought of as a vectorbundle over the circle (since the unit interval(0, 1) is homeomorphic to R). This vector bundle is distinct from the trivial bundle E = R1 ×S 1 , but both Whitney sums E ⊕ E and M ⊕ Mare equivalent to the trivial bundle R2 × S 1 .whole numberwosetA non-negative integer.See well-ordered set.Zorn’s LemmaZZermelo hierarchychy.See cumulative hierar-Zermelo set theoryZermelo-Fraenkel settheory without the Axiom of Replacement. Abbreviated by the letter Z. See Zermelo-Fraenkelset theory.Zermelo-Fraenkel set theoryThe formal theory whose axioms are: the Axiom ofExtensionality, the Axiom of Regularity, the Axiom of Pairing, the Axiom of Separation, theAxiom of Union, the Axiom of Power Set, theAxiom of Infinity, the Axiom of Replacement,and the Axiom of Choice. This axiomatic theory is often abbreviated as ZFC (the letter C isfor the Axiom of Choice).zero Symbol: 0(1) A symbol representing the absence ofquantity.(2) The additive identity of an Abelian groupA.

The element denoted as 0 ∈ A which hasthe property that 0 + a = a + 0 = a for everyelement a ∈ A.© 2001 by CRC Press LLCzero object An object A of a category C thatis both terminal and initial is a zero object ofC. Such an object is usually denoted by 0 or ∗,and is also called a null object of the category.For example, in the category of Abelian groupsand group homomorphisms, ({0}, +) is a zeroobject. Any two zero objects are isomorphic.zero sectionA map M −→ E of a vectorbundle E −→ M over a manifold M, whichtakes each point m in M to the zero in the vectorspace which is the fiber over m. That this mapis well defined follows from the definition ofvector bundle.Example: For any trivial bundle M × Rn ,M × {0} is the zero section.The term zero section can also refer to theimage of the section map.ZF Zermelo-Fraenkel set theory without theAxiom of Choice.

See Zermelo-Fraenkel settheory.ZFCSee Zermelo-Fraenkel set theory.Zorn’s Lemma If (P, ≤) is a nonempty partial order in which every chain has an upperbound, then P has a maximal element. In otherwords, if for every linearly ordered C ⊆ P thereis a pc ∈ P such that q ≤ pc for all q ∈ C,then there is one p ∈ P such that q ≤ p forall q ∈ P. Zorn’s Lemma is equivalent to theAxiom of Choice..

Характеристики

Список файлов книги