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A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1] This equivalent condition is formally expressed as follows:
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.
However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on intersection theory, [1] uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X×Y such that Z is finite and surjective over each ...
For example, if is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of . The orthonormal frame bundle is just a reduction of the structure group of F G L ( M ) {\displaystyle F_{\mathrm {GL} }(M)} to the orthogonal group O ( n ) {\displaystyle \mathrm {O} (n)} .
A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. [ 1 ] [ 2 ] Set-valued functions are used in a variety of mathematical fields, including optimization , control theory and game theory .
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...