But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.

In many cases such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin.

In that case it is equivalent to saying that if we have several ( a finite number of ) boxes, each containing at least one item, then we can choose exactly one item from each box.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups.

Even in this case, however, it is still possible to list all the elements, because the set is finite ; it has a specific number of elements.

The differential equations determining the evolution function Φ < sup > t </ sup > are often ordinary differential equations: in this case the phase space M is a finite dimensional manifold.

In the case where V is of finite dimension n it is common to choose a basis for V and identify GL ( V ) with GL ( n, K ) the group of n-by-n invertible matrices on the field K.

In this case, the group is also called a permutation group ( especially if the set is finite or not a vector space ) or transformation group ( especially if the set is a vector space and the group acts like linear transformations of the set ).

* The nominative case indicates the subject of a finite verb: We went to the store.

* The nominative case ( subjective pronouns such as I, he, she, we ), used for the subject of a finite verb and sometimes for the complement of a copula.

proved the theorem ( for the special case of polynomial rings over a field ) in the course of his proof of finite generation of rings of invariants.

This might come in the form of a proof that the number in question is in fact irrational ( or rational, as the case may be ); or a finite algorithm that could determine whether the number is rational or not.

The simplest way to define infinite dimensional Lie groups is to model them on Banach spaces, and in this case much of the basic theory is similar to that of finite dimensional Lie groups.

In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite dimensional Lie groups no longer hold.

In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median ( in the case of an even sample size ), and, if there is such a member, there may be more than one so that the median may not uniquely identify a sample member.

* In the case of PSK ( phase-shift keying ), a finite number of phases are used.

* In the case of FSK ( frequency-shift keying ), a finite number of frequencies are used.

* In the case of ASK ( amplitude-shift keying ), a finite number of amplitudes are used.

* In the case of QAM ( quadrature amplitude modulation ), a finite number of at least two phases, and at least two amplitudes are used.

Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classic case of the natural numbers ordered by divisibility ; for an account of those, see incidence algebra.

These algorithms are not practicable for hand written computation, but are available in any Computer algebra system ( see Berlekamp's algorithm for the case in which the coefficients belong to a finite field or the Berlekamp – Zassenhaus algorithm when working over the rational numbers ).

The condition on the fundamental group turns out to be necessary ( and sufficient ) for finite time extinction, and in particular includes the case of trivial fundamental group.

In the digital QAM case, a finite number of at least two phases and at least two amplitudes are used.

Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired ( as is often the case in category theory ).

note that the above online algorithm III is a special case of an algorithm that works for any partition of the sample into sets,:

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

But note that in each case, the statute sets the general principles, but the common law process determines the scope and application of the statute.

In the case of infinite sets, the behavior is more complex.

In this case two sets of strings can be stored, one for display purposes, and another for collation purposes.

Reloading dies and shell holders for 7. 5mm Swiss Dies are generally sold in sets of two or three dies, depending on the shape of the case.

* Size / Resize the case ( For previously fired cases, primer is pressed out in this step with most die sets.

If the two sets of bodies do not have concurrent jurisdiction but, as in the case of the International Criminal Court ( ICC ), the relationship is expressly based on the principle of complementarity, i. e. the international court is subsidiary or complementary to national courts, the difficulty is avoided.

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum ( as is the case with the arithmetic mean ) e. g. rates of growth.

Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra.

He then sets up a dialogue in which the disputant offers a series of ways of defining S, and he meets each with a suitable objection, so drawing the conclusion that in such a case there is no right definition of S.

Some scuba sets incorporate attached extra stage cylinders, as bailout in case the main breathing gas supply is used up or malfunctions, or containing another gas mixture.

The < tt > LC_COLLATE = C </ tt > sets the default collation order to not fold upper and lower case together, and the < tt >"$@"</ tt > causes any parameters given to < tt > l </ tt > to be passed through as parameters to ls, so that all of the normal options and other syntax known to ls can still be used.

In that case, the theist appears to face a dilemma: either to accept that both sets of responses are equally bad, and so that the theist does not have an adequate response to the problem of evil ; or to accept that both responses are equally good, and so to consider the existence of an omnipotent, omniscient, and omnimalevolent being as plausible.

The Chief Justice sets the agenda for the weekly meetings where the justices review the petitions for certiorari, to decide whether to hear or deny each case.

In many cases, the special case of thermal noise arises, which sets a fundamental lower limit to what can be measured or signaled and is related to basic physical processes described by thermodynamics, some of which are expressible by simple formulae.

Ur-elements can be treated as a different logical type from sets ; in this case, makes no sense if is an ur-element, so the axiom of extensionality simply applies only to sets.

ZPP is the class of sets for which any random string is likely to be a witness of x in X, or x not in X, which ever the case may be.