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...For the ring of rational integers '''Z''' one may also take ''S'' to be a finite set of prime numbers and define an ''S''-unit to be an integer divisible only b

...gonality is not assured. However, it is always possible to orthogonalize a finite set of linearly independent set of vectors, i.e., to form linear superpositions

...t of [[real number]]s with the [[counting measure]], which assigns to each finite set of reals the number of points in the set. This measure space is not ''σ''

...either an upper or a lower bound. A ''directed set'' is one in which any finite set has an upper bound.

...The machine is always in one 'state' which is an element in a predefined, finite set of states. The head is also always in a single location on the tape (i.e.

For a finite set of data

For a finite set of data

On a finite set ''S'' of ''n'' elements, consider a function σ that defines for each

.... A related property is that rational numbers are the only numbers with [[finite set|finite]] expansions as [[continued fraction|regular continued fractions]].

If ''S'' is a [[finite set]] of numbers, then all its elements can be added together using sigma notat

...number of elements are called infinite sets. The number of elements that a finite set contains is called that set's [[cardinality]]. The concept of cardinality c

...ents in <math>A.</math> This case is called the uniform distribution (on a finite set <math>\Omega</math>), or a symmetric probability space. The uniform distrib

Any permutation of a finite set can be written as a product of permutations called ''cycles''. A cycle <mat

...finite'' if it has finitely many elements, that is if the set ''G'' is a [[finite set]].

...ive grammar is formally defined as one that is fully ''explicit''. It is a finite set of rules that can be applied to ''generate'' exactly those sentences (often

...t this in fact holds for any set (Cantor's Theorem). This is obvious for a finite set, but Cantor's ingenious proof made no reference to the set being finite; th

...there is always another prime number which is not in that set. Choose any finite set of prime numbers <math>\scriptstyle \{ p_1, p_2, p_3, \ldots, p_n \}</math> ...t. Since any finite set of prime numbers can thus be extended to a larger finite set of prime numbers, we conclude that there are infinitely many prime numbers.

# In mathematics, the number of elements in a finite set. See [[Exponent|exponent]].

...t this in fact holds for any set (Cantor's Theorem). This is obvious for a finite set, but Cantor's ingenious proof made no reference to the set being finite; th

...hed between about 1955 and 1983, which is thought to classify all of the [[finite set|finite]] [[simple group]]s into roughly 30 basic types.

..., there is always another prime number that is not in that set. Choose any finite set of prime numbers <math>\scriptstyle \{ p_1, p_2, p_3, \ldots, p_n \}</math> ...t. Since any finite set of prime numbers can thus be extended to a larger finite set of prime numbers, we conclude that there are infinitely many prime numbers.

...language to be an infinite collection of sentences that is generated by a finite set of rules from finite [[lexicon|lexical]] items. Crucially, these sentences

...'', where ''X'' is the finite set of all possible inputs, and ''Y'' is the finite set of all possible outputs. Existence of this function ''f'' is called "counte ...ts are provided. The input ''x''_A is an element of a prescribed finite set ''X''_A (not necessarily a number); the same holds for ''y''<sub>

...to the way in which an infinite number of curves can be drawn through any finite set of data points on a graph. This thesis was accepted by [[Karl Popper]], lea