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 production rules ( left-hand side right-hand side ) where each side consists of a sequence of these symbols

* 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?

* A finite set N of nonterminal symbols.

* A finite set N of nonterminal symbols.

The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed ; frequently it is required to be finite.

* The alpha set is a finite set of elements called proposition symbols or propositional variables.

* The omega set is a finite set of elements called operator symbols or logical connectives.

The changes in the carrier signal are chosen from a finite number of M alternative symbols ( the modulation alphabet ).

* is a finite set of symbols ( the tape alphabet )

A nested stack automaton allows full access, and also allows stacked values to be entire sub-stacks rather than just single finite symbols.

In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set called an alphabet.

In mathematics, the symmetric group S < sub > n </ sub > on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself.

* is a finite, non-empty set of the tape alphabet / symbols

* Word, a finite string of symbols in automata theory

In Reed – Solomon coding, source symbols are viewed as coefficients of a polynomial p ( x ) over a finite field.

The original concept of Reed – Solomon coding describes encoding of k message symbols by viewing them as coefficients of a polynomial p ( x ) of maximum degree k − 1 over a finite field of order N, and evaluating the polynomial at n > k distinct input points.

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

The input of the problem consists of two finite lists and of words over some alphabet having at least two symbols.

At any time, the symbols so far fed to the automaton as input form a finite sequence of symbols, which is called a word.

Since all computational problems are reducible into the accept / reject question on words ( all problem instances can be represented in a finite length of symbols ),

:* Σ is a finite set of symbols, called the alphabet of the automaton.

: An automaton reads a finite string of symbols a < sub > 1 </ sub >, a < sub > 2 </ sub >,...., a < sub > n </ sub >, where a < sub > i </ sub > ∈ Σ, which is called an input word.

The first scientific theories indicating that the age of the Universe might be finite were the studies of thermodynamics, formalized in the mid 19th century.

Players would usually be given a finite number of lives ( or attempts ) to progress through the game which when expended would usually result in the display of the message " Game Over " indicating that the game had ended.