Although passengers rarely have cause to refer to either document, all travel is subject to the National Rail Conditions of Carriage and all tickets are valid subject to the rules set out in a number of so-called technical manuals, which are centrally produced for the network.

A prefix code is a code with the " prefix property ": there is no valid code word in the system that is a prefix ( start ) of any other valid code word in the set.

A set of house rules may be accepted as valid by a group of players wherever they play, as it may also be accepted as governing all play within a particular house, café, or club.

ICANN has approved the Internationalizing Domain Names in Applications ( IDNA ) system, which maps Unicode strings into the valid DNS character set using Punycode.

Climacus rejects Hegel's suppression of individuality by pointing out it is impossible to create a valid set of rules or system in any society which can adequately describe existence for any one individual.

A valid argument is a set of propositions, the last of which follows from — or is implied by — the rest.

The simplest valid argument is modus ponens, one instance of which is the following set of propositions:

When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters.

Thermodynamics states a set of four laws which are valid for all systems that fall within the constraints implied by each.

This set is defined in the HTML 4. 0 DTD, which also establishes the syntax ( allowable sequences of characters ) that can produce a valid HTML document.

Searle's rebuttal suggests that " one obvious difficulty with it arguments against hierarchical ranking of books is that if it were valid, it would argue against any set of required readings whatever ; indeed, any list you care to make about anything automatically creates two categories, those that are on the list and those that are not.

It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles.

ICANN approved the Internationalized domain name ( IDNA ) system, which maps Unicode strings used in application user interfaces into the valid DNS character set by an encoding called Punycode.

We define Thm ( C ) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod ( X ) be the class of all frames which validate every formula from X.

In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general.

Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non-existence of a set of all sets may or may not remain valid.

In mathematics, logic and computer science, a formal language is called recursively enumerable ( also recognizable, partially decidable or Turing-acceptable ) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i. e., if there exists a Turing machine which will enumerate all valid strings of the language.

Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a formal proof from a finite set of axioms is valid.

Congressional leaders tried to resolve the crisis by creating a 15-member Electoral Commission that would determine which set of votes were valid.

Under this principle, an explanation is valid if it is the best possible explanation of a set of known data.

Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

In his work Semantography Bliss had not provided a systematic set of definitions for his symbols ( there was a provisional vocabulary index instead ( 1965, pp. 827 – 67 )), so McNaughton ’ s team might often interpret a certain symbol in a way that Bliss would later criticize as a “ misinterpretation ”.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

Several studies have suggested that hedge funds are sufficiently diversifying to merit inclusion in investor portfolios, but this is disputed for example by Mark Kritzman who performed a mean-variance optimization calculation on an opportunity set that consisted of a stock index fund, a bond index fund, and ten hypothetical hedge funds.

* The ring of p-adic integers is the inverse limit of the rings Z / p < sup > n </ sup > Z ( see modular arithmetic ) with the index set being the natural numbers with the usual order, and the morphisms being " take remainder ".

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

A few new instructions were introduced with the 80186 ( referred to as the 8086-2 instruction set in some datasheets ): enter / leave ( replacing several instructions when handling stack frames ), pusha / popa ( push / pop all general registers ), bound ( check array index against bounds ), and ins / outs ( input / output of string ).

* Diversity index – a quantitative measure that increases when the number of types into which a set of entities has been classified increases, and obtains its maximum value for a given number of types when all types are represented by the same number of entities

Internally, the HC11 instruction set is upward compatible with the 6800, with the addition of a Y index register.

This is done as follows: if I is some index set and R < sub > i </ sub > is a ring for every i in I, then the cartesian product Π < sub > i in I </ sub > R < sub > i </ sub > can be turned into a ring by defining the operations coordinatewise, i. e.

The coproduct is given by the disjoint union: given sets A < sub > i </ sub > where i ranges over some index set I, we construct the coproduct as the union of A < sub > i </ sub >×

Meta search engines store neither an index nor a cache and instead simply reuse the index or results of one or more other search engines to provide an aggregated, final set of results.

An alternative method is to estimate the current altitude ( angle ) of the sun from navigation tables, then set the index bar to that angle on the arc, apply suitable shades only to the index mirror, and point the instrument directly at the horizon, sweeping it from side to side until a flash of the sun's rays are seen in the telescope.

; Index error: This occurs when the index and horizon mirrors are not parallel to each other when the index arm is set to zero.

Each set of synonyms ( synset ), has a unique index and shares its properties, such as a gloss ( or dictionary ) definition.

Some researchers allow infinitary operations, such as where J is an infinite index set, thus leading into the algebraic theory of complete lattices.

When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility index encode a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences.

If I is an index set and X < sub > I </ sub > is the set of indeterminates X < sub > i </ sub > for i ∈ I, then a monomial X < sup > α </ sup > is any finite product of elements of X < sub > I </ sub > ( repetitions allowed ); a formal power series in X < sub > I </ sub > with coefficients in a ring R is determined by any mapping from the set of monomials X < sup > α </ sup > to a corresponding coefficient c < sub > α </ sub >, and is denoted.

These relations consist of a heading and a set of zero or more tuples in arbitrary order.

The tuples are a set of unique attributes mapped to values, or the rows of data in the table.

* Extension ( predicate logic ), the set of tuples of values that satisfy the predicate

: A relation is a tuple with, the header, and, the body, a set of tuples that all have the domain.

* Relation ( database ), a set of tuples in a relational database

The distinction between tuples in the relational model and those in set theory is only superficial ; the above example can be interpreted as a 2-tuple if an arbitrary total order is imposed on the attributes ( e. g. ) and then the elements are distinguished by this ordering rather than by the attributes themselves.

In the relational model, a relation is a ( possibly empty ) finite set of tuples all having the same finite set of attributes.

You must be careful to avoid a mismatch that may arise between the two languages because negation, applied to a formula of the calculus, constructs a formula that may be true on an infinite set of possible tuples, while the difference operator of relational algebra always returns a finite result.

The Cartesian product is defined differently from the one in set theory in the sense that tuples are considered to be ' shallow ' for the purposes of the operation.

A relation is a set of tuples.

The set of all tuples over D is denoted as T < sub > D </ sub >.

The result of the query is the set of tuples X < sub > i </ sub > to X < sub > n </ sub > which makes the DRC formula true.

The tuples ( A, B, t ) where A is an invertible complex n by n matrix, B is any complex n by n matrix, and t is any complex number from an open set in complex space of dimension 2n < sup > 2 </ sup > + 1.

For functional programming, it provides several constructs and a set of immutable types: tuples, records, discriminated unions and lists.

To see the connection with the classical picture, note that for any set S of polynomials ( over an algebraically closed field ), it follows from Hilbert's Nullstellensatz that the points of V ( S ) ( in the old sense ) are exactly the tuples ( a < sub > 1 </ sub >, ..., a < sub > n </ sub >) such that ( x < sub > 1 </ sub >-a < sub > 1 </ sub >, ..., x < sub > n </ sub >-a < sub > n </ sub >) contains S ; moreover, these are maximal ideals and by the " weak " Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.

# the relation does not have two distinct tuples ( i. e. rows or records in common database language ) with the same values for these attributes ( which means that the set of attributes is a superkey )

Since a relation contains no duplicate tuples, the set of all its attributes is a superkey if NULL values are not used.

The set of tuples that are greater than or equal to some particular tuple forms a positive orthant with its apex at the given tuple.

* In every infinite set of-tuples of natural numbers, there exist two tuples and such that, for every,.

The tuples in correspond one-for-one with the monomials over a set of variables.

Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences ( though mathematics does speak of ordered sets, and these can be seen to include the musical kind in some sense, they are far more involved ).

More precisely, find necessary and sufficient conditions on the tuple ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) and ( y < sub > 1 </ sub >, ..., y < sub > n </ sub >) separately, so that there is an element of R with the property that x < sub > i </ sub >· r = y < sub > i </ sub > for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.

The row is then interpreted as a relvar composed of a set of tuples, with each tuple consisting of the two items: the name of the relevant column and the value this row provides for that column.