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A set S of natural numbers is called recursively enumerable if there is a partial recursive function ( synonymously, a partial computable function ) whose domain is exactly S, meaning that the function is defined if and only if its input is a member of S.
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set and S
In practical situations there will be restrictions on the admissible operating conditions, and we regard the vectors as belonging to a fixed and bounded set S.
The set of vectors Af constitutes the operating policy or, more briefly, the policy, and a policy is admissible if all the Af belong to S.
The stage was then set for the campaign for statewide election of the Illinois legislature which would, in turn, select Lincoln or Douglas as its U. S. senator.
Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.
** For every non-empty set S there is a binary operation defined on S that makes it a group.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
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 >).
* If a set S is in M and S is congruent to T then T is also in M and a ( S )
* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )
set and natural
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.
set and 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.
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.
set and is
`` 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.
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