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.

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

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

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 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.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

Unlike McCaffrey's black crystal transceivers, Le Guin's ansibles are not mated pairs: it is possible for an ansible's coordinates to be set to any known location of a receiving ansible.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.

The latter notation corresponds to viewing R as the characteristic function on " X " x " Y " for the set of pairs of G.

Molecules are typically a set of atoms bound together by covalent bonds, such that the structure is electrically neutral and all valence electrons are paired with other electrons either in bonds or in lone pairs.

Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all possible pairs of real numbers ; that is with the Cartesian product, where is the set of all reals.

* The relation " is a sibling of " ( used to connote pairs of distinct people who have the same parents ) on the set of all human beings is not an equivalence relation.

The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes all pairs of elements related is the coarsest.

* Let be the set of ordered pairs of integers with not zero, and define an equivalence relation on according to which if and only if.

A groupoid is a set G with a unary operation and a partial function Here * is not a binary operation because it is not necessarily defined for all possible pairs of G-elements.

This principle is widely used in computer graphics, computational geometry and many other disciplines, to solve many proximity problems in the plane or in three-dimensional space, such as finding closest pairs in a set of points, similar shapes in a list of shapes, similar images in an image database, and so on.

In both cases, it is assumed that the training set consists of a sample of independent and identically distributed pairs,.

In supervised learning, we are given a set of example pairs and the aim is to find a function in the allowed class of functions that matches the examples.

If S contains two elements that are not pairwise orthogonal ( in particular, the set of all quantum states includes such pairs ) then an argument like that given above shows that the answer is no.

The set of all ordered pairs whose first entry is in some set X and whose second entry is in some set Y is called the Cartesian product of X and Y, and written X × Y.

The 32 tiles in a Chinese dominoes set can be arranged into 16 pairs, as shown in the picture at the top of this article.

To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices.

Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs ( partition, partial order ).

For a b, the interval is the set of points x satisfying a x and x b, also written a x b. It contains at least the points a and b. One may choose to extend the definition to all pairs ( a, b ).

* If a player has no pairs, straights or flushes, he should set the second-and third-highest cards in his two-card hand.

Following shots would be in pairs, one to set off the reactive armor, the second to penetrate the tank's armor.

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.

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.

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.