For example, the set of rational numbers — those numbers which can be written as a quotient of integers — contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.

In the general case of a differentiable bijection, the concept of scale can, to some extent, still be used, but it may depend on location and direction.

With every bijection from the space to itself two coordinate transformations can be associated:

Because a bijection can only exist between two sets when they have the same size, the number of elements in F depends only on the size of A.

For locally small categories, this can be expressed by the existence of a bijection between the hom-sets

A set has cardinality if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or " one-to-one correspondence ", with the natural numbers.

This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality ( a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above ; in the style of Hilbert's Hotel Infinity ).

Such a proof was given by Désiré André, based on the observation that the unfavourable sequences can be divided into two equally probable cases, one of which ( the case where B receives the first vote ) is easily computed ; he proves the equality by an explicit bijection.

Because is a bijection, semigroup actions can be defined as functions which satisfies

A set S is called countable if there exists an injective function f from S to the natural numbers Since there is an obvious bijection between and it makes no difference whether one considers 0 to be a natural number of not.

In order for a compression algorithm to be considered lossless, there needs to exist a reverse mapping from compressed bit sequences to original bit sequences ; that is to say, the compression method would need to encapsulate a bijection between " plain " and " compressed " bit sequences.

But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating ( T is a partial function ), so this fails to produce the required bijection.

A bijection g ( t ) from T to ( 0, 1 ) is defined by: If t is the n < sup > th </ sup > string in sequence b, let g ( t ) be the n < sup > th </ sup > number in sequence a ; otherwise, let g ( t ) = 0. t.

Thus, if one class is " small enough " to be a set, and there is a bijection from that class to a second class, the axiom states that the second class is also a set.

( Sketch of proof: let A, B, C be the sets of residue classes modulo-and-coprime-to m, n, mn respectively ; then there is a bijection between A × B and C, by the Chinese remainder theorem.

In an ideal phonemic orthography, there would be a complete one-to-one correspondence ( bijection ) between the graphemes ( letters ) and the phonemes of the language, and each phoneme would invariably be represented by its corresponding grapheme.

Alice " blinds " the message by encoding it into some other input E ( x ); the encoding E must be a bijection on the input space of f, ideally a random permutation.

The two ways of counting S must each be by establishing a bijection of S with a set obviously counted by the number found.

However its numerator counts the Cartesian product of k finite sets of sizes n,, ...,, while its denominator counts the permutations of a k-element set ( the set most obviously counted by the denominator would be another Cartesian product k finite sets ; if desired one could map permutations to that set by an explicit bijection ).

More abstractly and generally, we note that the two quantities asserted to be equal count the subsets of size k and n − k, respectively, of any n-element set S. There is a simple bijection between the two families F < sub > k </ sub > and F < sub > n − k </ sub > of subsets of S: it associates every k-element subset with its complement, which contains precisely the remaining n − k elements of S. Since F < sub > k </ sub > and F < sub > n − k </ sub > have the same number of elements, the corresponding binomial coefficients must be equal.

This definition of " infinite set " should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form

Sets A and B of natural numbers are said to be recursively isomorphic if there is a total computable bijection f from the set of natural numbers to itself such that f ( A ) = B.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

Two sets are said to have the same cardinality or cardinal number if there exists a bijection ( a one-to-one correspondence ) between them.

What we have done here is arranged the integers and the odd integers into a one-to-one correspondence ( or bijection ), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

Two sets have the same cardinal number if and only if there is a bijection between them.

In mathematical analysis, an isomorphism between two Hilbert spaces is a bijection preserving addition, scalar multiplication, and inner product.

And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number which doesn't fit, and thereby proves a contradiction.

This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros.

There is a bijection between the field orderings of F and the positive cones of F.

There is a bijection between every pair of equivalence classes: the inverse of a representative of the first equivalence class, composed with a representative of the second.

This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F

To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable.

There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle.

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements, and thus isomorphic, meaning that there is a bijection between them, for example

If φ is a bijection between sets A and B, then F is a bijection between the sets of F-structures F and F, called transport of F-structures along φ.

By the above observation, the sequences form a partition of the whole of the disjoint union of A and B, hence it suffices to produce a bijection between the elements of A and B in each of the sequences separately.

For an A-stopper, the function ' is a bijection between its elements in A and its elements in B.

For a B-stopper, the function ' is a bijection between its elements in B and its elements in A.

We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform.

Formally, a set S is called finite if there exists a bijection

For example, the " species of permutations " maps each finite set A to the set of all permutations of A, and each bijection from A to another set B naturally induces a bijection from the set of all permutations of A to the set of all permutations of B.

This establishes a bijection ( up to isomorphism ) between the class of all finite posets and the class of all finite distributive lattices.

For the group algebra of a finite group, the ( isomorphism types of ) projective indecomposable modules are in a one-to-one correspondence with the ( isomorphism types of ) simple modules: the socle of each projective indecomposable is simple ( and isomorphic to the top ), and this affords the bijection, as non-isomorphic projective indecomposables have

Bijective numeration is any numeral system that establishes a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits.