For the only time in the opera, words are not set according to their natural inflection ; ;

The difficulty appears when there is no natural choice of elements from each set.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

The standard structure is where is the set of natural numbers, is the successor function and is naturally interpreted as the number 0.

* The game Brink is set on a futuristic arcology to preserve humanity after a natural flooding disaster.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."

It seems natural that a set of individuals ought to exist, so long as the individuals exist.

Division of whole numbers can be thought of as a function ; if Z is the set of integers, N < sup >+</ sup > is the set of natural numbers ( except for zero ), and Q is the set of rational numbers, then division is a binary function from Z and N < sup >+</ sup > to Q.

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set

In mathematics, a countable set is a set with the same cardinality ( number of elements ) as some subset of the set of natural numbers.

The elements of a countable set can be counted one at a time — although the counting may never finish, every element of the set will eventually be associated with a natural number.

Some authors use countable set to mean a set with the same cardinality as the set of natural numbers.

A set S is called countable if there exists an injective function f from S to the natural numbers Since there is an obvious bijection between and it makes no difference whether one considers 0 to be a natural number of not.

Some sets are infinite ; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size.

For example, Georg Cantor ( who introduced this concept ) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers ( non-negative integers ), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

* The set of algebraic numbers is countable ( enumerable ).

* Hence, the set of algebraic numbers has Lebesgue measure zero ( as a subset of the complex numbers ), i. e. " almost all " complex numbers are not algebraic.

* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

Each orbital is defined by a different set of quantum numbers ( n, l, and m ), and contains a maximum of two electrons each with their own spin quantum number.

These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i. e., by occupation schemes of atomic orbitals ( e. g., 1s < sup > 2 </ sup > 2s < sup > 2 </ sup > 2p < sup > 6 </ sup > for the ground state of neon -- term symbol: < sup > 1 </ sup > S < sub > 0 </ sub >).

Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers.

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

Some large telephone companies have toll-free numbers set up.

These numbers are set up by a company offering low charge calls in the UK, these numbers are meant to be used as a sort of operator that you go through in order to qualify for these cheap calls.

I was at least conscious of the distinction in my full Yokuts presentation that awaits publication, in which, in listing ' Two-Stem Meanings ', I set off by asterisks those forms in which N of stem B was Af of stem A/3, the unasterisked ones standing for Af ; ;

or under ' Four Stems ', I set off by asterisks cases where the combined N of stems Af was Af.

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

The Boltzmann distribution for the fractional number of particles N < sub > i </ sub > / N occupying a set of states i possessing energy E < sub > i </ sub > is:

S < sub > N </ sub > 1 leads to the non-stereospecific addition and does not result in a chiral center, but rather in a set of geometric isomers ( cis / trans ).

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.

Each of the columns from 3 to N is set to a value derived from the first and higher derivatives of the polynomial.

Morgan filled the fire box with wood again, then started supper and set the table.

Clapping spurs to the bronc he set off at a sharp canter, with growing alarm.

Her face was pale but set and her dark eyes smoldered with blame for Ben.

They recognized that slavery was a moral issue and not merely an economic interest, and that to recognize it explicitly in their Constitution would be in explosive contradiction to the concept of sovereignty they had set forth in the Declaration of 1776 that `` all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among them are life, liberty and the pursuit of happiness.

Opposite every gate was a hitching post or a stone carriage-step, set with a rusty iron ring for tying a horse.

The wife, Amra, and her lover are both savagely portrayed, she as incarnate sensuality, `` voluptuous '' and `` indolent '', possibly `` a mischief maker '', with `` a kind of luxurious cunning '' to set against her apparent simplicity, her `` birdlike brain ''.

We will recall that the still confident liberals of the Truman administration gathered with other Western utopians in San Francisco to set up the legal framework, finally and at last, to rationalize war -- to rationalize want and fear -- out of the world: the United Nations.

So when textbooks, like that of Baker set out drawings of the ' Ptolemaic System ', complete with earth in the center and the seven heavenly bodies epicyclically arranged on their several deferents, we have nothing but a misleading 20th-century idea of what never existed historically.

The strongest appeal of the Copernican formulation consisted in just this: ideally, the justification for dealing with special problems in particular ways is completely set out in the basic ' rules ' of the theory.

You probably would not remember, since you never seemed to remember even the same moments as I, much less their intensity, one sunny midday on Fifth Avenue when you had set out with me for some final shopping less than a week before the wedding you staged for me with such reluctance at the Farm.

The work had its beginning in 1938 with an eight-bar musical strain to which Koehler set the words `` There'll be no more work ; ;

An entire theater had been set up for his diversion, with a 200-man Italian orchestra under the well-known Sarti.

Criticism is as old as literary art and we can set the stage for our study of three moderns if we see how certain critics in the past have dealt with the ethical aspects of literature.

He could produce carefully constructed orations, set and formal speeches, artfully and prayerfully made by writing and rewriting with all the aid his tutor and others could provide, and then delivered verbatim from memory.

Among some recent imports were seat covers for one series of dining room chairs on which were depicted salad plates overflowing with tomatoes and greens and another set on which a pineapple was worked in naturalistic color.

In particular he sought the gentle, sweet-faced nuns, with head coverings and veils coming to the middle of their foreheads, remembering their expressions until he reached home and set them down on paper.

She filled a big pitcher and set it, with glasses, on a tray.

The purchase was effected and they made their way towards the hotel again, the hen, with whom some sort of communication had been set up, nestling in the doctor's arms.

It is equipped with electronic controls that can be set to hold precise tension and speed.

the purpose of producing plays at the College is three-fold: to provide the Carleton students with the best possible opportunity for theater-going within the limits set by the maturity and experience of the performers and the theatrical facilities available ; ;

Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.

The Rio Grande KC is also considering having their Junior Classes set up so that Juniors can qualify with points for Westminster.

He is a fine-looking colt with a good body, good set of legs and nice way of going.

This one was set up here in 390 A.D. on a pedestal, the faces of which are carved with statues of the emperor and his family watching games in the Hippodrome, done so realistically that the obelisk itself is included in them.