The graph, as a set, may have a finite number of components.

In the C-plane we construct a set of rectangular Cartesian coordinates u, V with the origin at Q and such that both C and Af have finite slope at Q.

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.

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

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

Clearly, the set of even numbers is infinitely large ; there is no requirement that a set be finite.

* a finite set of nonterminal symbols ( indicating that some production rule can yet be applied )

* a finite set of terminal symbols ( indicating that no production rule can be applied )

Even in this case, however, it is still possible to list all the elements, because the set is finite ; it has a specific number of elements.

A set is countable if: ( 1 ) it is finite, or ( 2 ) it has the same cardinality ( size ) as the set of natural numbers.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

* Any finite topological space, including the empty set, is compact.

Similarly, the set of rational numbers in the closed interval is not compact: the sets of rational numbers in the intervals and cover all the rationals in for but this cover does not have a finite subcover.

* The set R of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover.

The cardinality of a finite set is a natural number – the number of elements in the set.

It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

An example of an NP-complete problem is the subset sum problem: given a finite set of integers is there a non-empty subset which sums to zero?

What if Generally Accepted Accounting Principles recognized natural and human capital not as a free amenity in inexhaustible supply but as a finite and integrally valuable factor of production?

He claimed that private ownership of the means of production is theft, and that the finite nature of the earth imposes moral restrictions on the right to acquire unlimited property.

The production lines for rifled 120 mm ammunition in the UK have been closed for some years, so existing stocks of ammunition for the L30A1 are finite.

OPT was billed as the first software to provide finite capacity scheduling for production environments.

One of his main arguments in Small is Beautiful is that we cannot consider the problem of technological production solved if it requires that we recklessly erode our finite natural capital and deprive future generations of its benefits.

However, natural gas is a finite resource like all fossil fuels, and production of natural gas is expected to peak soon after oil.

* A finite set P of production rules, where a rule has one of the following forms:

Advanced CAx tools merge many different aspects of the product lifecycle management ( PLM ), including design, finite element analysis ( FEA ), manufacturing, production planning, product

A W-grammar consists of a finite set of meta-rules, which are used to derive ( possibly infinitely many ) production rules from a finite set of hyper-rules.

It consists of a finite number of integral production plans.

Syntax was recognized as the focal point of language production, in which a finite set of rules can produce an infinite number of sentences.

The simulation considers population, employment, basic water management and ecology, goods ( availability and production ), raw materials ( ore, steel, coal ), services ( education, health, fire protection, leisures ), energy ( electricity and charcoal, coal with finite reserves, solar and wind power ) and other constraints such as finance, pollution and transports.

The problem requires the production of n functions of the variable z, regular throughout the complex z-plane except at the given singular points ; at these points the functions may become infinite of only finite order, and when z describes circuits about these points the functions shall undergo the prescribed linear substitutions.

However, as signaled by abundant particle production present in strong interactions, the quark structure of strongly interacting particles allows an infinite number of " degrees of freedom " to be present in finite volume.

Analogously, it may ( or may not ) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.

Chomsky frequently quotes Humboldt's description of language as a system which " makes infinite use of finite means ", meaning that an infinite number of sentences can be created using a finite number of grammatical rules.

Cell division has been modeled by finite subdivision rules.

By this analogy, it is suggested, the experience of free will emerges from the interaction of finite rules and deterministic parameters that generate nearly infinite and practically unpredictable behaviourial responses.

* Expression ( mathematics ), a finite combination of symbols that are well-formed according to applicable rules

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Instead of phrase structure rules ATNs used an equivalent set of finite state automata that were called recursively.

Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the " Hydra " is a rooted tree, and a move consists of cutting off one of its " heads " ( a branch of the tree ), to which the hydra responds by growing a finite number of new heads according to certain rules.

From the point of view of formal languages this is immediately a problem, since syntax rules are expected to generate finite objects.

The system thus consists of any number of formulas built up through finite combinations of the primitive symbols — combinations that are formed from the axioms in accordance with the stated rules.

), give a finite number of propositions expressed in those symbols, which were to be taken as " foundations " ( the axioms ), and some rules of inference which would model the way humans make conclusions.

The inference engine can be described as a form of finite state machine with a cycle consisting of three action states: match rules, select rules, and execute rules.

Here X is a finite set of " points " and Y is a class of subsets of X, called " blocks ", subject to rules that depend on the type of design.

" ( Actually the rules for determining a leap year are more complex than that ; but there is a finite number of rules, and you could in principle include them all in the sentence.

It is a finite set of rules that can be applied to generate all those and only those sentences ( often, but not necessarily, infinite in number ) that are grammatical in a given language.

In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

A key feature of language is that a simple, finite set of phonological items gives rise to an infinite lexical system wherein rules determine the form of each item, and meaning is inextricably linked with form.

* A finite set P of parsing rules.

Generative science shows how deterministic and finite rules and parameters in the natural phenomena interact with each other to generate seemingly unanticipated and infinite behaviour.