The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop.

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

This representation exhibits the set of all invertible affine transformations as the semidirect product of K < sup > n </ sup > and GL ( n, k ).

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.

In a monoid, the set of ( left and right ) invertible elements is a group, called the group of units of, and denoted by or H < sub > 1 </ sub >.

A square matrix with entries in a field is invertible ( in the set of all square matrices of the same size, under matrix multiplication ) if and only if its determinant is different from zero.

Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.

In mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation of ordinary matrix multiplication.

For example, the general linear group over R ( the set of real numbers ) is the group of n × n invertible matrices of real numbers, and is denoted by GL < sub > n </ sub >( R ) or GL ( n, R ).

More generally, the general linear group of degree n over any field F ( such as the complex numbers ), or a ring R ( such as the ring of integers ), is the set of n × n invertible matrices with entries from F ( or R ), again with matrix multiplication as the group operation.

The set of all invertible diagonal matrices forms a subgroup of GL ( n, F ) isomorphic to ( F < sup >×</ sup >)< sup > n </ sup >.

Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).

For example, supposing there are K basic symbols, an alternative Gödel numbering could be constructed by invertibly mapping this set of symbols ( through, say, an invertible function h ) to the set of digits of a bijective base-K numeral system.

The tuples ( A, B, t ) where A is an invertible complex n by n matrix, B is any complex n by n matrix, and t is any complex number from an open set in complex space of dimension 2n < sup > 2 </ sup > + 1.

On the other hand, if the non-zero entries are only required to be non-zero, but not necessarily invertible, this set of matrices forms a semigroup instead.

A set of matrices are said to be simultaneously diagonalisable if there exists a single invertible matrix P such that is a diagonal matrix for every A in the set.

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

A set of tables containing spectral intensities for 39,000 lines of 70 elements, as observed in a copper matrix in a d-c arc, was completed and published.

This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor ; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family.

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

* In set theory, an automorphism of a set X is an arbitrary permutation of the elements of X.

Similarly, the subset order ⊆ on the subsets of any given set is antisymmetric: given two sets A and B, if every element in A also is in B and every element in B is also in A, then A and B must contain all the same elements and therefore be equal:

The repeating periodicity of the blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons which occupy a complete set of s, p, d and f atomic orbitals, respectively.

Another set of methods is the aromatization of cyclohexanes and other aliphatic rings: reagents are catalysts used in hydrogenation such as platinum, palladium and nickel ( reverse hydrogenation ), quinones and the elements sulfur and selenium.

* Consider a set with three elements, A, B, and C. The following operation:

The set of valid index tuples and the addresses of the elements ( and hence the element addressing formula ) are usually, but not always, fixed while the array is in use.

In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.

An abelian group is a set, A, together with an operation "•" that combines any two elements a and b to form another element denoted.

A partially ordered set ( poset ) P is said to satisfy the ascending chain condition ( ACC ) if every ascending chain of elements eventually terminates.

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

In mathematics, a binary operation on a set is a calculation involving two elements of the set ( called operands ) and producing another element of the set ( more formally, an operation whose arity is two ).

More precisely, a binary operation on a non-empty set S is a map which sends elements of the Cartesian product S × S to S:

* Base ( group theory ), a sequence of distinct elements of a set

Bartók and Kodály quickly set about incorporating elements of such Magyar peasant music into their compositions.

That mistake, she thought, had cost her dearly these past few days, and she wanted to avoid falling into any more of the traps that the mountain might set for her.

And his performances attracted much attention, as the frequency of his surviving pieces in any calendar that may be set up for his undergraduate activities testifies.

A fire had just been lighted, he saw, and things had been set out for drinks, and, like any stray, his response to these comforts was instantaneous.

In the event the rupees set aside for loans under Section 104 ( E ) of the Act are not advanced within six years from the date of this Agreement because the Export-Import Bank of Washington has not approved loans or because proposed loans have not been mutually agreeable to the Export-Import Bank of Washington and the Department of Economic Affairs of the Government of India, the Government of the United States of America may use the rupees for any purpose authorized by Section 104 of the Act.

One year I simply set the plants in the remains of a compost pile, to which a little sand had been added, and I had the most beautiful pansies in my, or any of my neighbors' experience.

The depth gauge rod can now be set, and any number of holes bored to exact and identical depth.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

The opportunity exists for states to reserve some of their vocational education funds to apply on an ad hoc flexible basis to subsidize any local preemployment training programs that may be quickly set up in a community to aid a new industrial plant.

At any rate, Manchester did not lag far behind the first commercial system which was set up in 1844 between Baltimore and Washington.

Personally, it is the Southerner's sexual coming of age, when, without any warning, unbreakable taboos are set up between himself and his past.

Gun set the captain's fifth of Hiram Walker inside the safe before he reported to Lt. Killpath, though he knew that Killpath's ulcer prevented him from making any untoward incursion on Herman Wolff's gift.

The presiding judge also decides at what amount, if any, to set bail.

: For any set X of nonempty sets, there exists a choice function f defined on X.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

: For any set A, the power set of A ( with the empty set removed ) has a choice function.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.