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.

In the case of finite sets, this agrees with the intuitive notion of size.

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.

The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.

( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

# X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover ( Alexander's sub-base theorem )

A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense.

Whereas we say X is an H-set of Z if every cover of X with open sets of Z has a finite subfamily whose Z closure contains X.

The complementary problem is in co-NP and asks: " given a finite set of integers, does every non-empty subset have a nonzero sum?

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.

So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component.

A directed set is a set A with a preorder such that every finite subset of A has an upper bound.

Intuitively, an expander is a finite, undirected multigraph in which every subset of the vertices " which is not too large " has a " large " boundary.

The asks for an algorithm that takes as input a statement of a first-order logic ( possibly with a finite number of axioms beyond the usual axioms of first-order logic ) and answers " Yes " or " No " according to whether the statement is universally valid, i. e., valid in every structure satisfying the axioms.

The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs ( possibly with an uncountable number of vertices ) for which every finite subgraph is planar.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

Wedderburn's little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field.

* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.

The coupling of this corollary with the initial statement of the law proves every threaded discussion to be finite in length.