Bijection, injection and surjection

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.A function is injective (one-to-one) if or, equivalently, if One could also say that elements of the codomain (sometimes called range by mistake) are mapped to by at most one element (argument) of the domain; not every element of the codomain, however, need have an argument mapped to it. An injective function is an injection.A function is surjective (onto) if every element of the codomain is mapped to by some element (argument) of the domain; this is expressed logically by saying that for all in , Note that with this definition, some images may be mapped to by more than one argument. (Equivalently, a function where the range is equal to the codomain.) A surjective function is a surjection.A function is bijective (one-to-one and onto) if and only if (iff) it is both injective and surjective. (Equivalently, every element of the codomain is mapped to by exactly one element of the domain.) A bijective function is a bijection (one-to-one correspondence).
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In mathematics and related technical fields, the term map or mapping is often a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, with the same meaning.
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