Note that if " n " > " x " then one obtains a factor zero, so in this case there are no injective functions at all; this is just a restatement of the pigeonhole principle.
22.
Formally, X \ preceq ^ { PD } Y if-and-only-if there exists an Injective function f from X to Y such that, for each x \ in X, x \ preceq f ( x ).
23.
An injective function need not be surjective ( not all elements of the codomain may be associated with arguments ), and a surjective function need not be injective ( some images may be associated with " more than one " argument ).
24.
This is in contrast to data type constructors, which define injective functions from all types of a particular kind to a new set of types, and type synonyms ( a . k . a . typedef ), which define functions from all types of a particular kind to another existing set of types using a single case.
25.
Hello RD . I am aware that any analytic injective function on the whole complex plane ( i . e . entire, injective ), must be of the form az + b, a ! = 0-I just found a very nice proof using the Casorate-Weierstrass theorem, much more concise than my previous efforts.
26.
An admissible ordinal \ alpha is called " nonprojectible " if there is no total \ alpha-recursive injective function mapping \ alpha into a smaller ordinal . ( This is trivially true for regular cardinals; however, we are mainly interested in countable ordinals . ) Being nonprojectible is a much stronger condition than being admissible, recursively inaccessible, or even recursively Mahlo.
27.
By the fact that " f " and " g " are injective functions, each " a " in " A " and " b " in " B " is in exactly one such sequence to within identity : if an element occurs in two sequences, all elements to the left and to the right must be the same in both, by the definition of the sequences.
28.
If f : D \ to A and B \ subseteq A, then saying that the image of f is B is equivalent to saying that f with its codomain restricted to B is surjective . ( I doubt that this is the problem, but you do know, don't you, that a " one-to-one function " is another name for an " injective function ", but a " one-to-one correspondence " is another name for a " bijective function " . ) talk ) 18 : 05, 23 February 2010 ( UTC)
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