Compound comparisons typically compare two sets of groups means where one set has two or more groups ( e. g., compare average group means of group A, B and C with group D ).

But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.

It contains 2800 proposed cognate sets, a set of sound laws based on those proposed sets, and a number of grammatical correspondences, as well as a few important changes to the reconstruction of Proto-Altaic.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

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

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

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.

: 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.

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 example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

After enough attacks, he decides to take the evil head-on equipped with a shotgun and a chainsaw and sets out to face the enemy.

* G-sets, sets equipped with an action of a group G.

Let ( X, Σ ) and ( Y, Τ ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras Σ and Τ.

* An internationalized system is equipped for use in a range of " locales " ( or by users of multiple languages ), by allowing the co-existence of several languages and character sets for input, display, and UI.

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets.

Later, RDF sets were equipped with rotatable ferrite loopstick antennae, which made the sets more portable and less bulky.

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

After ČKD bankruptcy in 2000 Alstom ( as new owner of FIAT Ferroviaria ) took over whole order, reducing number of sets to seven and raising total price, while provided an improved version of ETR 470, equipped to work on, and 3, 000 V DC, thus suitable to operate in Germany, Austria, Slovakia and the Czech Republic.

Dom sets out, with Hrsh-Hgn ( his tutor, a swamp-dwelling phnobe ), Isaac ( his robot, equipped with Man-Friday subcircuitry ) and Ig ( his pet swamp ig ) in tow, on a picaresque adventure to find the Jokers ' world.

* The FCC stated that, as of March 1, 2007, all televisions must be equipped with digital tuners, but stores that had TV sets with analog tuners only could continue to sell analog-tuner TV sets.

All access to the structure interior must be equipped with two sets of doors or revolving door ( airlock ).

The Brown Gymnasium has an NCAA regulation-sized basketball court, with several sets of bleachers and a sky box equipped for events, as well as a workout and fitness center for athletes.

Finally, equipped with his father's consciousness duplicated in a Mother Box, and joined by Mikaal Tomas, Jack sets off on an intergalactic journey.

A group scheme is a group object in a category of schemes that has fiber products and some final object S. That is, it is an S-scheme G equipped with one of the equivalent sets of data

Album sets of the past were arranged " in sequence " for phonographs equipped with record changers.

This was not so much of a format as the RF option was an adaptation necessary for plugging in such devices on television sets that did not come equipped with a composite video input.

* February 27 – Piloting a United States Army Air Service Packard-Le Peré LUSAC-11 fighter equipped with one of the first turbochargers, Major Rudolf Schroeder sets a new world altitude record of.

* September 6 – Flying the Wright XF3W-1 Apache equipped with floats, United States Navy Lieutenant Apollo Soucek sets a world altitude record for seaplanes, climbing to 38, 500 feet ( 11, 735 meters ).

An extensive modernization drive has ensured that these vehicles are equipped with wireless sets in communication with a central control room.

After rebuilding, the R46s were renumbered to 5482-6258, equipped with LCD side signs, lost their MTA blue stripe paint-scheme, and arranged in linked sets of four.

Numerous special versions of Zaporozhets were equipped with additional sets of controls that allowed operating the car with a limited set of limbs or completely by hands, and were given for free or with big discounts to the disabled people, especially war veterans, as an alternative to SMZ-series microcars ( such variants reached at times up to 20 % of ZAZ output ).

Much more dangerous were Mosquito intruders equipped with a device called Serrate to allow them to track German night fighters by emissions from their Lichtenstein B / C, C-1 or SN-2 sets.

Fifty lantern slide teaching sets on the subject of `` Emergency War Surgery ( NATO ) '' were assembled and distributed to the Medical Military Services of foreign Governments associated with NATO and South-East Asia Treaty Organization.

The demand for teaching sets continues unabated since they provide the means for the military physicians to review the pathology of selected disease processes or organ systems for review of basic sciences and correlation of clinical physiological behavior with structural changes.

Both home and auto radios were in excellent demand, with retail sales of home sets ahead of 1959 in every month of the first eleven ; ;

The last exercise of Roland Claude's prescribed program for Henri is a single exercise, done in individual sets with a bit longer pause between sets.

So with four complete Push-Pull Super-Sets No. 1, four of No. 2, four of No. 3 and four to six sets of the Incline Bench Press, you can see that Henri De Courcy has had a terrific mass-building, muscle-shaping, torso-defining workout that cannot be improved upon.

Squat-style lifters and leg-split lifters would both benefit enormously by practicing those variations providing that they remember to make alternate sets with the left and right leg to the front.

In the frame are two sets of bars which interact with each other to prevent the operator from making dangerous moves.

It was `` window '' -- the tinsel paper dropped by bombers to jam radar sets, to fill the scope with hundreds of blips that would seem to be approaching bombers.

The sets are remarkably elaborate for a road-show that doesn't pause long in any one place, and they are devised so that they shift with a minimum of interruption or obtrusiveness.

So the plot creaks, the sets are decaying, the costumes are pre-historic, the orchestra was sloppy and not very well connected with what the singers were doing.

These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

The same notation is used with sets to denote cardinality ; the meaning depends on context.

In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.

Articles of faith are sets of beliefs usually found in creeds, sometimes numbered, and often beginning with " We believe ...", which attempt to more or less define the fundamental theology of a given religion, and especially in the Christian Church.

In contrast, in the Arian German kingdoms established on the wreckage of the Western Roman Empire in the 5th century, there were entirely separate Arian and Nicene Churches with parallel hierarchies, each serving different sets of believers.