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One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them.
The function g : R → R defined by g(x) = x 2 is not surjective, since there is no real number x such that x 2 = −1. However, the function g : R → R ≥0 defined by g(x) = x 2 (with the restricted codomain) is surjective, since for every y in the nonnegative real codomain Y, there is at least one x in the real domain X such that x 2 = y.
The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective. The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...), up to the number of elements in the counted set. It results that two finite sets ...
In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
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One-to-one (communication), the act of an individual communicating with another; One-to-one (data model), a relationship in a data model; One to one computing (education), an initiative for a computer for every student; One-to-one marketing or personalized marketing, an attempt to make a unique product offering for each customer
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