Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Well-ordering theorem: Every set can be well-ordered.

Every information exchange between living organisms — i. e. transmission of signals that involve a living sender and receiver can be considered a form of communication ; and even primitive creatures such as corals are competent to communicate.

Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form.

* Every regular language is context-free because it can be described by a context-free grammar.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form.

Every real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every entire function can be represented as a power series that converges uniformly on compact sets.

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

Every sequence can, thus, be read in three reading frames, each of which will produce a different amino acid sequence ( in the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every species can be given a unique ( and, one hopes, stable ) name, as compared with common names that are often neither unique nor consistent from place to place and language to language.

Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.

Every morpheme can be classified as either free or bound.

Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication.

Every document window is an object with which the user can work.

Every adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and can make arrangements for the care of his / her dependants during the trip, must perform the Hajj once in a lifetime.

Every ordered field can be embedded into the surreal numbers.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

Every First Family seems to have one couple upon whom it relies for true friendship.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.

1 ) Every age rewrites the events of its history in terms of what should have been, creating legends about itself that rationalize contemporary beliefs and excuse contemporary actions.

Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.

Every member of the family must have a vital place in its life.

Every single instance of altruistic behavior need not always increase inclusive fitness ; altruistic behaviors would have been selected for if such behaviors on average increased inclusive fitness in the ancestral environment.

Disraeli wrote a personal letter to Gladstone, asking him to place the good of the party above personal animosity: " Every man performs his office, and there is a Power, greater than ourselves, that disposes of all this ..." In responding to Disraeli Gladstone denied that personal feelings played any role in his decision then and previously to accept office, while acknowledging that there were differences between him and Derby " broader than you may have supposed.

Every one of us has been passing through days of anxiety ; we cannot, however, feel that peace has been established, but that we have nothing but an armistice in a state of war.

Every time you have a Prime Minister who wants to make all the decisions, it mainly leads to bad results.

Every adult and wise woman without husband although reserves the right to accept concubinage for her employer / master without any coercion or force, but Islamic scholars have consensus that for sex relations options of Nikkah and Muta should be adopted instead of concubinage in this era.

Most of his films have been at least partially financed by Telefilm Canada, and Cronenberg is a vocal supporter of government-backed film projects, saying " Every country needs system of government Grant ( money ) | grants in order to have a national cinema in the face of Hollywood ".

Every connected graph is an expander ; however, different connected graphs have different expansion parameters.

Every system must have a class designated as " root ", with one of its creation procedures designated as " root procedure ".

Every four years, during which an extra 24 hours have accumulated, one extra day is added to keep the count coordinated with the sun's apparent position.

Cobham was a common butt of veiled satire in Elizabethan popular literature ; he figures in Ben Jonson's Every Man in His Humour and may have been part of the reason The Isle of Dogs was suppressed.

Every line of serious work that I have written since 1936 has been written directly or indirectly against totalitarianism and for Democratic Socialism as I understand it.

" Orwell stated in " Why I Write " ( 1946 ): " Every line of serious work that I have written since 1936 has been written, directly or indirectly, against totalitarianism and for democratic socialism, as I understand it.

::: Every Bill which shall have passed the House of Representatives and the Senate, shall, before it become a Law, be presented to the President of the United States ; If he approves he shall sign it, but if not he shall return it, with his Objections to that House in which it shall have originated ...

Every Mac made between 1986 and 1998 has a SCSI port on the back, making external expansion easy ; also, " toaster " Compact Macs did not have easily accessible hard drive bays ( or, in the case of the Mac Plus, any hard drive bay at all ), so on those models, external SCSI disks were the only reasonable option.

Every space filling curve hits some points multiple times, and does not have a continuous inverse.

This was to be based on utilitarian principles ; he said: " Every man has a right to that, the exclusive possession of which being awarded to him, a greater sum of benefit or pleasure will result than could have arisen from its being otherwise appropriated.

Every now and again wulin needs to have a champion, a general or a commander to lead the collective resources of wulin participants for China.

When coming to Kubla Khan, he pointed out: " instead of being content to have written finely under the influence of laudanum, recommends ' Kubla-Khan ' to his readers, not as a poem, but as ' a psychological curiosity ' ... Every lover of books, scholar or not, who knows what it is to have his quarto open against a loaf at his tea ... ought to be in possession of Mr. Coleridge's poems, if it is only for ' Christabel ', ' Kubla Khan ', and the ' Ancient Mariner '.