Completeness : Complete
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion.
- Metric spaces or uniform spaces are said to be complete if every Cauchy sequence in them converges. See Complete space.
- In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. In the particular case of Hilbert spaces (or more generally, inner product spaces), an orthonormal basis is a set that is both complete and orthonormal.
- A measure space is complete if every subset of every null set is measurable. See Complete measure.
- In statistics, a statistic is called complete if it doesn't allow an unbiased estimator of zero. See Completeness (statistics).
- In graph theory, a complete graph is an undirected graph where every pair of vertices has exactly one edge connecting them.
- In category theory, a category C is called complete if every functor from a small category to C has a limit; it is called cocomplete if every such functor has a colimit.
- In logic, a formal calculus (often just specified by a set of additional axioms used to formalize some theory within the underlying logic) is said to be complete if, for any statement P, a proof exists for P or for not P. A system is consistent if a proof never exists for both P and not P. Gödel's incompleteness theorem proved that no system as powerful as the Peano axioms can be both consistent and complete. See also below for another notion of completeness in logic.
- In proof theory and related fields of mathematical logic, a formal calculus is said to be complete with respect to a certain logic (i.e. wrt its semantics), if every statement P, that follows sematically from a set of premisses G, can be derived syntactically from these premisses within the calculus. Formally, G|=P implies G|-P. Especially, all tautologies of the logic can be proven. Even when working with classical logic, this isn't equivalent to the notion of completeness introduced above (both a statement and its negation might not be tautologies wrt the logic). The reverse implication is called soundness.
- In complexity theory, a problem P is said to be complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-Complete is complete for the class NP, under polynomial-time, many-one reduction.
And then the King and the Cid
vassals, and told them how King Don Alfonso had banished him from
and who would remain at home. Then Alvar Fanez, who was his cousin-
through desert and through peopled country, and never fail you. In
parments, and ever while we live be unto you loyal friends and
Cid thanked them for their love, and said that there might come a
when he saw his hall deserted, the household chests unfastened, the
upon the perches, the tears came into his eyes, and he said, "My
turned toward the East and knelt and said, "Holy Mary Mother, and
destroy all the Pagans, and to win enough from them to requite my
called for Alvar Fanez and said unto him, "Cousin, the poor have no
be done unto them along our road," and he called for his horse.
My Cid Ruydiez entered Burgos, having sixty streamers in his
Burgos and the women of Burgos were at their windows, weeping, so
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Alfonso in his anger had sent letters to Burgos, saying that no man
lose all that he had, and moreover the eyes in his head. Great
when he came near them because they did not dare speak to him; and
fastened, for fear of the King. And his people called out with a
the door, and took his foot out of the stirrup, and gave it a kick,
girl of nine years old then came out of one of the houses and said
not open our doors to you, for we should lose our houses and all
you, but God and all His saints be with you." And when she had.
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