The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.

A pair of amicable numbers is a set of sociable numbers of order 2.

An important conjecture due to Catalan with respect to aliquot sequences is that every aliquot sequence ends in one of the above ways – with a prime number, a perfect number, or a set of amicable or sociable 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.

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

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

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.

`` All I have to do to set the record is to go on down.

And the best way to conceal and disguise the elements of an incest story is not to set out to write an incest story.

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.

Once the scene is set, Trevelyan skilfully builds up the tense story until it reaches its climax in the dramatic victory of Marlborough and Eugene of Savoy at Blenheim.

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.

To say this is to set aside the realness of time.

I refer to the notion that the structure of society is a microcosm of the cosmic design and that history conforms to patterns of justice and chastisement as if it were a morality play set in motion by the gods for our instruction.

Even to be `` from hope and fear set free '' is at least better than to have lost the first without having got rid of the second.

I believe it is an area in which professional planners have failed to set adequate guide posts ; ;

It is high time the United States began to realize that the God-given rights of men set forth in that document are applicable today to Katanga.

One of the inescapable realities of the Cold War is that it has thrust upon the West a wholly new and historically unique set of moral dilemmas.

The day is now appropriately set aside to honor the American men and women who have contributed to the success of our merchant marine fleet in peace and war.

The Act further provides for a `` floor '' or minimum allotment, set at the 1954 level, which is called the `` base '' allotment, and a `` ceiling '' or maximum allotment, for each State.

The method used in computing the allotments is specifically set forth in the Act.

The method used for computing the respective Federal and State shares in total program costs is specifically set forth in the Act.

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 ; ;

The first substantially complete stereo Giselle ( and the only one of its scope since Feyer's four-sided LP edition of 1958 for Angel ), this set is, I'm afraid, likely to provide more horrid fascination than enjoyment.

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

There is no limit to what you can spend, yet it is easily possible to keep within a set budget.

The main set of bars are the `` tappets '' and one tappet is connected to each lever.

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.

There is even one set that has `` barbecue '' written on it.