And, yes, the vacuously celebratory lyrics hyping the "swell celebrity" Jimmy Walker may raise a wan smile.
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Working with the screenwriter Seth W. Owen, and expanding some of the ideas in his 2012 short film, "Loom," Mr. Scott creates a disappointingly skeletal Frankenstein story that entertains efficiently but vacuously.
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Because she said what she thought, and because she smiled only when she felt like smiling, and not constantly and vacuously, America's cheapest caricature was cast on her: the Angry Black Woman.
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Each cycle through r_1 then r_2 therefore either doubles the initial number of Ss, or converts the Ss to as. The trivial case of generating a, in case it is difficult to see, simply involves vacuously applying r_1, thus jumping straight to r_2 which also vacuously applies, then jumping to r_3 which produces a.
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In case x has no accessible worlds then R(x,y) is false but the whole formula is vacuously true: an implication is also true when the antecedent is false.
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In category theory, the empty semigroup is always admitted. It is the unique initial object of the category of semigroups. A semigroup with no element is an inverse semigroup, since the necessary condition is vacuously satisfied.
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The empty set can be considered a derangement of itself, because it has only one permutation (0!=1), and it is vacuously true that no element (of the empty set) can be found that retains its original position.
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In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction.. This notion has relevance in pure mathematics, as well as in any other field that uses classical logic. Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with.
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In mathematics and logic, a vacuous truth is a conditional or universal statement that is only true because the antecedent cannot be satisfied. For example, the statement "all cell phones in the room are turned off" will be true even if there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on and turned off". For that reason, it is sometimes said that a statement is vacuously true only because it does not really say anything.
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P A Grillet (1995). Semigroups. CRC Press. pp. 3–4 One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from to S. This operation vacuously satisfies the closure and associativity axioms of a semigroup.
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The following conditions are equivalent for a poset P: #P is a disjoint union of zigzag posets. #If a ≤ b ≤ c in P, either a = b or b = c. #< \circ < = \emptyset, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
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We use induction on . If is empty, then the theorem is vacuously true and the base case for induction is verified. Assume is non-empty, let be an element of and write If is any -linear transformation on , by the induction hypothesis there exists such that for all in . Write .
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Thus any intersection between f_1 and f_2 cannot be transverse. However, non-intersecting manifolds vacuously satisfy the condition, so can be said to intersect transversely. #When \ell_1 + \ell_2 = m, the image of L_1 and L_2's tangent spaces must sum directly to M's tangent space at any point of intersection.
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Similarly, a right Euclidean relation is right unique if, and only if, it is anti- symmetric. # A left Euclidean and left unique relation is vacuously transitive, and so is a right Euclidean and right unique relation. # A left Euclidean relation is left quasi-reflexive. For left-unique relations, the converse also holds.
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Sixteen dollars for a glorified platform is preposterous when you can get all of the information in the book—most of which means next to nothing—for free online. It begins vacuously (its first two sentences are: 'It has been said that America is great because America is good. We agree.') and doesn't get better from there.
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In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism).Some authors define a theory to be categorical if all of its models are isomorphic. This definition makes the inconsistent theory categorical, since it has no models and therefore vacuously meets the criterion. Such a theory can be viewed as defining its model, uniquely characterizing its structure.
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Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but showed that for all sufficiently large N these series have zeros with real part greater than . Therefore, Turán's result is vacuously true and cannot help prove the Riemann hypothesis.
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The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the Krull dimension of the ring, we know that it should be 1.
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The empty set is the set containing no elements. In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.
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The empty set is a subset of every set (the statement that all elements of the empty set are also members of any set A is vacuously true). The set of all subsets of a given set A is called the power set of A and is denoted by 2^A or P(A); the "P" is sometimes in a script font. If the set A has n elements, then P(A) will have 2^n elements.
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Every member of the empty set is a nonempty set (that is vacuously true), and their union is the empty set. Therefore, the empty set is the only partition of itself. As suggested by the set notation above, we consider neither the order of the partitions nor the order of elements within each partition. This means that the following partitionings are all considered identical: :{ {b}, {a, c} } :{ {a, c}, {b} } :{ {b}, {c, a} } :{ {c, a}, {b} }.
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He also pursued a notable career as an organist and choir-trainer. After serving as organist-choirmaster of St Mary's Roman Catholic Church in Clapham, he was Master of Music of Westminster Cathedral for 12 years (1947–1959). He developed the choir's forthright, full-throated tone—often, but rather vacuously described as "continental"—which contrasted with that of Anglican choirs at the time. Benjamin Britten praised the choir's 'staggering brilliance and authority', and proposed to write a piece for them.
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According to the material conditional analysis, a natural language conditional, a statement of the form ‘if P then Q’, is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true. Goodman illustrates this point using the following pair in a context where it is understood that the piece of butter under discussion had not been heated.Goodman, N., "The Problem of Counterfactual Conditionals", The Journal of Philosophy, Vol.
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Language controlled grammars are grammars in which the production sequences constitute a well-defined language of arbitrary nature, usually though not necessarily regular, over a set of (again usually though not necessarily) context-free production rules. They also often have a sixth set in the grammar tuple, making it G = (N, T, S, P, R, F), where F is a set of productions that are allowed to apply vacuously. This version of language controlled grammars, ones with what is called "appearance checking", is the one henceforth.
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In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard algorithmic complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NPThe unique games conjecture is vacuously true if P = NP, as then every problem in NP would also be NP-hard.
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An alternate argument for the principle stems from model theory. A sentence P is a semantic consequence of a set of sentences \Gamma only if every model of \Gamma is a model of P. However, there is no model of the contradictory set (P \wedge \lnot P). A fortiori, there is no model of (P \wedge \lnot P) that is not a model of Q. Thus, vacuously, every model of (P \wedge \lnot P) is a model of Q. Thus Q is a semantic consequence of (P \wedge \lnot P).
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In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first-order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of (the codomain of) a forgetful functor, and in physics of (hypothesized) hidden variable theories purporting to explain quantum mechanics. Elementary classes are (vacuously) pseudoelementary but the converse is not always true; nevertheless pseudoelementary classes share some of the properties of elementary classes such as being closed under ultraproducts.
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The constant presheaf with value Z, which we will denote F, is the presheaf that chooses all four sets to be Z, the integers, and all restriction maps to be the identity. F is a functor, hence a presheaf, because it is constant. F satisfies the gluing axiom, but it is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets: Vacuously, any two sections of F over the empty set are equal when restricted to any set in the empty family.
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A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology. Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (under the subspace topology). Note however, that if a space were called locally normal if and only if each point of the space belonged to a subset of the space that was normal under the subspace topology, then every topological space would be locally normal. This is because, the singleton {x} is vacuously normal and contains x.
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They dubbed a song that sounded curiously like the former "One More Song on the Market", while "Nothing to Say in a House Song" railed against the duo's perceived vacuousness in house music by (vacuously) repeating the title with no other lyrics. "Shampoo Victims" also included a duo of collaborations with friends/heroes Sparks. Russell and Ron Mael provide both lyrics and vocals on the album's best tracks "La nuit est là" (The Night Is Here) and "Yo quiero mas dinero" (I Want More Money). As if to accentuate this fondness for '70s/'80s pop, the duo also sampled Giorgio Moroder's "Love Fever" on "Each Finger Has an Attitude".
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In mathematics, a non-Desarguesian plane, named after Girard Desargues, is a projective plane that does not satisfy Desargues' theorem, or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is valid in all projective spaces of dimension not 2,Desargues' theorem is vacuously true in dimension 1; it is only problematic in dimension 2. that is, all the classical projective geometries over a field (or division ring), but David Hilbert found that some projective planes do not satisfy it. . According to the footnote on this page, the original "first" example appearing in earlier editions was replaced by Moulton's simpler example in later editions.
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Either the CNF formula Φ is found to comprise a consistent set of literals--that is, there is no `l` and `¬l` for any literal `l` in the formula. If this is the case, the variables can be trivially satisfied by setting them to the respective polarity of the encompassing literal in the valuation. Otherwise, when the formula contains an empty clause, the clause is vacuously false because a disjunction requires at least one member that is true for the overall set to be true. In this case, the existence of such a clause implies that the formula (evaluated as a conjunction of all clauses) cannot evaluate to true and must be unsatisfiable.
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Theoretical Computer Science 1989.. To see this, first notice that the inclusion TFNP ⊆ F(NP \cap coNP) follows easily from the definitions of the classes. All "yes" answers to problems in TFNP can be easily verified by definition, and since problems in TFNP are total, there are no "no" answers, so it is vacuously true that "no" answers can be easily verified. For the reverse inclusion, let R be a binary relation in F(NP \cap coNP). Decompose R into R1\cup R2 such that (x, 0y) ∈ R1 precisely when (x, y) ∈ R and y is a "yes" answer, and let R2 be (x, 1y) such (x, y) ∈ R and y is a "no" answer.
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If the terms 'p', 'q' and 'r' stand for arbitrary propositions then the main paradoxes are given formally as follows: # These, which are all equivalent to (p \lor eg p) \lor q : ##( eg p \land p) \to q, p and its negation imply q. This is the paradox of entailment. ## eg p \to (p \to q) or p \to (q \lor p), if p is false then it implies every q, in which cases the statement p \to q is said to be vacuously true; or if p is true then it implies itself or every q since p \to p is equivalent to p \lor eg p. This is referred to as 'explosion'.
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The Albertson conjecture is vacuously true for n\le 4. In these cases, K_n has crossing number zero, so the conjecture states only that the n-chromatic graphs have crossing number greater than or equal to zero, something that is true of all graphs. The case n=5 of Albertson's conjecture is equivalent to the four color theorem, that any planar graph can be colored with four or fewer colors, for the only graphs requiring fewer crossings than the one crossing of K_5 are the planar graphs, and the conjecture implies that these should all be at most 4-chromatic. Through the efforts of several groups of authors the conjecture is now known to hold for all n\le 18.
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For n = 4, the h-cobordism theorem is true topologically (proved by Michael Freedman using a 4-dimensional Whitney trick) but is false PL and smoothly (as shown by Simon Donaldson). For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the 3-dimensional Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures. For n = 2, the h-cobordism theorem is equivalent to the Poincaré conjecture stated by Poincaré in 1904 (one of the Millennium Problems) and was proved by Grigori Perelman in a series of three papers in 2002 and 2003, where he follows Richard S. Hamilton's program using Ricci flow. For n = 1, the h-cobordism theorem is vacuously true, since there is no closed simply- connected 1-dimensional manifold.
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If constructive arithmetic is translated using realizability into a classical meta-theory that proves the \omega-consistency of the relevant classical theory (for example, Peano Arithmetic if we are studying Heyting arithmetic), then Markov's principle is justified: a realizer is the constant function that takes a realization that P is not everywhere false to the unbounded search that successively checks if P(0), P(1), P(2),\dots is true. If P is not everywhere false, then by \omega-consistency there must be a term for which P holds, and each term will be checked by the search eventually. If however P does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), P must either hold nowhere or not hold nowhere, therefore this constant function is a realizer.
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Zorn's lemma is sometimesFor example, , , and . stated as follows: Although this formulation appears to be formally weaker (since it places on P the additional condition of being non-empty, but obtains the same conclusion about P), in fact the two formulations are equivalent. To verify this, suppose first that P satisfies the condition that every chain in P has an upper bound in P. Then the empty subset of P is a chain, as it satisfies the definition vacuously; so the hypothesis implies that this subset must have an upper bound in P, and this upper bound shows that P is in fact non-empty. Conversely, if P is assumed to be non-empty and satisfies the hypothesis that every non-empty chain has an upper bound in P, then P also satisfies the condition that every chain has an upper bound, as an arbitrary element of P serves as an upper bound for the empty chain (that is, the empty subset viewed as a chain).
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A functor G : C → D induces a map from Cone(F) to Cone(GF): if Ψ is a cone from N to F then GΨ is a cone from GN to GF. The functor G is said to preserve the limits of F if (GL, Gφ) is a limit of GF whenever (L, φ) is a limit of F. (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.) A functor G is said to preserve all limits of shape J if it preserves the limits of all diagrams F : J → C. For example, one can say that G preserves products, equalizers, pullbacks, etc. A continuous functor is one that preserves all small limits. One can make analogous definitions for colimits. For instance, a functor G preserves the colimits of F if G(L, φ) is a colimit of GF whenever (L, φ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.
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