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.

Shānta-rasa functions as an equal member of the set of rasas but is simultaneously distinct being the most clear form of aesthetic bliss.

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

The term epískopos was not from the earliest times clearly distinguished from the term presbýteros (" elder ", " senior ", nowadays used to signify a priest ), but the term was already clearly used in the sense of the order or office of bishop, distinct from that of priest in the writings of Ignatius of Antioch ( died c. 108 ), and sources from the middle of the 2nd century undoubtedly set forth that all the chief centres of Christianity recognized and had the office of bishop, using a form of organization that remained universal until the Protestant Reformation.

For any set containing n elements, the number of distinct k-element subsets of it that can be formed ( the k-combinations of its elements ) is given by the binomial coefficient.

Other terms such as character set, character map, and code page are used almost interchangeably, but these terms have related but distinct meanings described below.

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

In particular, quantum phase transitions refer to transitions where the temperature is set to zero, and the phases of the system refer to distinct ground states of the Hamiltonian.

Immanuel Kant asserted that the world as we perceive it is organized according to a set of fundamental " intuitions ", which include object ( we perceive the world as a set of distinct things ); shape ; quality ( color, warmth, etc.

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.

Phonetic transcription ( representing each distinct speech sound with a separate symbol ) is shown with the International Phonetic Alphabet enclosed in square brackets, and phonemic transcription ( representing a small set of speech sounds that a particular language distinguishes ) is enclosed within virgules or slashes / /.

* 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.

When in doubt ( i. e. if a different name has not been clearly set in the literature ), the hierarchy expressed by the nomenclature should correspond to dynamically distinct ( sub -) systems in order of their dynamical relevance.

Hinduism is a conglomeration of distinct intellectual or philosophical points of view, rather than a rigid common set of beliefs, thus the basic definition of fundamentalism cannot apply to Hinduism considering that it does not contain any fundamental thoughts to abide to.

Hinduism is a conglomeration of distinct intellectual or philosophical points of view, rather than a rigid common set of beliefs.

A closely related but distinct character set is ISO-8859-1 ( distinguished by a dash after " ISO ").

Icehenge is part mystery, part psychological drama, and is set in three distinct time periods.

The UK is a unitary state formed initially by the merger of two independent kingdoms, the Kingdom of England and the Kingdom of Scotland, but the Treaty of Union ( 1707 ) that set out the agreed terms has ensured the continuation of distinct features of each state, including separate legal systems and separate national churches.

* Two-sided network effects: An increase in usage by one set of users increases the value to and participation of a complementary and distinct set of users, and vice versa.

The number of chromosomes in a single ( non-homologous ) set is called the monoploid number ( x ), and is distinct from the haploid number ( n ).

Moore, there can be no end to the process of set formation, and thus no such thing as the totality of all sets, or the set hierarchy.

Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.

Note that the set of grammars corresponding to recursive languages is not a member of this hierarchy.

At the third ( and in this case final ) layer, this set of choices is instantiated: which diseases will be included and in which branches of the hierarchy will they appear?

A hierarchy of committees was set up to improve discipline, transforming the party from an elite organisation to one of mass national appeal.

Murray was also criticised by Simpson for her " passionate system building ," developing her own image of a " rigidly codified and uniform system throughout Britain and Europe " that had its own set " Rites of initiation, dates for festivals, sabbath rituals, discipline and hierarchy within covens.

Under the U. S. legal system, courts are set up in a hierarchy.

: A judicial precedent attaches a specific legal consequence to a detailed set of facts in an adjudged case or judicial decision, which is then considered as furnishing the rule for the determination of a subsequent case involving identical or similar material facts and arising in the same court or a lower court in the judicial hierarchy.

Boniface again set out for what is now Germany, baptized thousands, and dealt with the problems of many other Christians who had fallen out of contact with the regular hierarchy of the Roman Catholic Church.

* The Cardinal ( 1963 ): a drama set in the Vatican hierarchy which earned Preminger his second Best Director Academy Award nomination

Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

A formal church hierarchy was set up, parallel to the civil administration, whose areas of responsibility often coincided.

Unless one can predict with certainty what qualities a set of objects and classes will have in the distant future, one cannot design a class hierarchy properly.

Press censorship was clarified and relaxed, some positions in the military hierarchy were made open to competition, and mutual schools were set up that encroached on the Catholic monopoly of public primary education.

Among the main features of RRG are the use of lexical decomposition, based upon the predicate semantics of David Dowty ( 1979 ), an analysis of clause structure, and the use of a set of thematic roles organized into a hierarchy in which the highest-ranking roles are ' Actor ' ( for the most active participant ) and ' Undergoer '.

A is optimal in its candidate set if it is better on the constraint hierarchy than all other candidates.

* Edmund Cooper's alternative-history The Cloud Walker is set in a world where the Luddite ethos has given rise to a religious hierarchy which dominates English society and sets carefully prescribed limits on technology.

For structured programming and procedural programming in general, great emphasis is placed on designing programs as a hierarchy of ( or at least a set of ) subroutines.

Joseph set about building a rationalized, centralized, and uniform government for his diverse lands, a hierarchy under himself as supreme autocrat.

) relating to an established hierarchy, procedure or set of specific behaviors.

The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.

Just as we can define what it means for a set X to be recursive relative to another set Y by allowing the computation defining X to consult Y as an oracle we can extend this notion to the whole arithmetic hierarchy and define what it means for X to be, or in Y, denoted respectively and.