For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

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, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.

For example, the set of all positive integers is infinite:

Thus intuitionists absolutely disallow the blanket assertion: " For all propositions P concerning infinite sets D: P or ~ P " ( Kleene 1952: 48 ).

For example, one could create a function that creates an infinite list ( often called a stream ) of Fibonacci numbers.

For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.

For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.

For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.

For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H = N, the center Z is contained in N, and so also in H, but then there is a smaller example H / Z whose normalizer in G / Z is N / Z = H / Z, creating an infinite descent.

For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values ( such as an interval of non-zero length ) may be positive.

For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types.

For a countably infinite set the set of possible order types is even uncountable.

For example, Stephen Kleene describes the notion of a Turing machine tape as " a linear tape, ( potentially ) infinite in both directions.

For the purposes of the Sprague – Grundy theorem, a game is a two-player game of perfect information satisfying the ending condition ( all games come to an end: there are no infinite lines of play ) and the normal play condition ( a player who cannot move loses ).

For example, the immortal monkey could randomly type G as its first letter, G as its second, and G as every single letter thereafter, producing an infinite string of Gs ; at no point must the monkey be " compelled " to type anything else.

For infinite sets, the supremum will always be 0 while the infimum can well be greater than 1.

For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy .< ref name =" Stillwell Infinite Series Early Results "> Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.

For it includes the emotional ties that bind men to their homeland and the complex motivations that hold a large group of people together as a unit.

For ten years a small group of European and U.S. critics has been calling attention to the half-forgotten Austrian expressionist Egon Schiele, who died 42 years ago at the age of 28.

For example, the steering committee might announce that the group felt a topic under study should not be dropped for an additional week as there was still too much of it untouched.

For purposes of sample selection only ( individual tests were given later ) we obtained group test scores of reading achievement and intelligence from school records of the entire third-grade population in each school system.

For if the small group notion involves the implicit claim that the phenomena of sociological investigations are of atomic or subatomic proportions, the philosopher needs to know the extent to which such entities are valid.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

For each element a of a group G, conjugation by a is the operation φ < sub > a </ sub >: G → G given by ( or a < sup >− 1 </ sup > ga ; usage varies ).

For example, three isomers exist for cresol because the methyl group and the hydroxyl group can be placed next to each other ( ortho ), one position removed from each other ( meta ), or two positions removed from each other ( para ).

For the bottom, only the group of higher priority need be considered.

For the U. S. Government's scientific enterprise, a significant impact of NAPAP were lessons learned in the assessment process and in environmental research management to a relatively large group of scientists, program managers and the public.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

For a few years, this group of animals was regarded as a subfamily, called the Callitrichinae, of the family Cebidae.

For example, it is thought that ammonites were the principal food of mosasaurs, a group of giant marine reptiles that became extinct at the boundary .< ref name =" Kauffman ">

For example, an " X " is used to indicate a variable group amongst a class of compounds ( though usually a halogen ), while " R " is used for a radical, meaning a compound structure such as a hydrocarbon chain.

For example, it is thought that ammonites were the principal food of mosasaurs, a group of giant marine reptiles that became extinct at the boundary .< ref name =" Kauffman ">

For example, it is immediately proven from the axioms that the identity element of a group is unique.

For the answer cannot be derived from any socially cohesive element in the disrupting community.

For example, " aloin " is derived from Aloe vera and Aloe ferox and has important medical uses ( e. g. as a laxative and in the treatment of burns ) as well as cosmetic uses ( e. g. in skin and hair products ).

For instance, according to Classical Arabic and Modern Standard Arabic, the Arabic root ( to sacrifice ) can be derived the forms ( he sacrificed ), ( you ( masculine singular ) sacrificed ), ( he slaughtered ), ( he slaughters ), and ( slaughterhouse ).

For each round, a subkey is derived from the main key using Rijndael's key schedule ; each subkey is the same size as the state.

For a long time, scholars believed that the alphorn had been derived from the Roman-Etruscan lituus, because of their resemblance in shape, and because of the word liti, meaning Alphorn in the dialect of Obwalden.

For commercial purposes, it is derived primarily from Gelidium amansii.

For instance, the name " Lord Sto Odin " in the story " Under Old Earth " is derived from the Russian words for " One hundred and one ", сто один.

For example, the CGS derived unit of force is the dyne, equal to, while the SI derived unit of force is the newton,.

For example, a physics lab course might ask students to record lengths in centimetres, and masses in grams, but force ( a derived unit ) in newtons, a usage consistent with the SI system.

For example, one might read that Corsican is a " central southern Italian dialect " along with Tuscan, Neapolitan, Sicilian and others or that it is " closely related to the Tuscan dialect of Italian ,", an infelicitous claim in that Italian is derived from Tuscan rather than the reverse.

For a finite group, the derived series terminates in a perfect group, which may or may not be trivial.

For example, using क ka, the following forms can be derived: क े ke, क ु ku, क ी kī, क ा kā, etc.

For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

For example butyl acetate ( systematically butyl ethanoate ), derived from butanol and acetic acid ( systematically ethanoic acid ) would be written CH < sub > 3 </ sub > CO < sub > 2 </ sub > C < sub > 4 </ sub > H < sub > 9 </ sub >.

For example, triphenyl phosphate is the ester derived from phosphoric acid and phenol.

For instance, the word photographer is derived from the word photograph by adding the derivational suffix -⟨ er ⟩.

For the name of the action, the suffix-ado will change a derived verb back to a noun: brosado ( a brushing ).

For instance, Wilfrid Sellars argued that non-doxastic mental states cannot be reasons, and so noninferential warrant cannot be derived from them.

For example: some early medieval manuscripts of Macrobius include maps of the Earth, including the antipodes, zonal maps showing the Ptolemaic climates derived from the concept of a spherical Earth and a diagram showing the Earth ( labeled as globus terrae, the sphere of the Earth ) at the center of the hierarchically ordered planetary spheres.

For example, the words preservative ( English ), préservatif ( French ), Präservativ ( German ), prezervativ ( Romanian, Czech, Croatian ), preservativ ( Slovenian ), preservativo ( Italian, Spanish, Portuguese ), prezerwatywa ( Polish ), презерватив " prezervativ " ( Russian, Serbian, Bulgarian ), prezervatif ( Turkish ), præservativ ( Danish ), prezervatyvas ( Lithuanian ), Prezervatīvs ( Latvian ) and preservatiu ( Catalan ) are all derived from the Latin word praeservativum.

For this reason, Gardnerian Wicca is usually considered to be the earliest extant tradition of Wicca, from which most subsequent Wiccan traditions are derived.

For instance, a granite which is formed from melted sediments may have more alkali feldspar, whereas a granite derived from melted basalt may be richer in plagioclase feldspar.