restricted to its image, and let be the inclusion map from into . Then . A dual factorization is given for surjections below. The composition of two injections is again an injection, but if is injective, then it can only be concluded that is injective (see figure). Every embedding is injective. Surjection A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain. In other words, each element of the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection. The formal definition is the following. The function is surjective, if for all , there is such that The following are some facts related to surjections: A function is surjective if and only if it is right-invertible, that is, if and only if there is a function such that identity function on . (This statement is equivalent to the axiom of choice.) By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection from a quotient set of its domain to its codomain. More precisely, the preimages under f of the elements of the image of are the equivalence classes of an equivalence relation on the domain of , such that x and y are equivalent if and only they have the same image under . As all elements of any one of these equivalence classes are mapped by on the same element of the codomain, this induces a bijection between the quotient set by this equivalence relation (the set of the equivalence classes) and the image of (which is its codomain when is surjective). Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of . The composition of two surjections is again a surjection, but if is surjective, then it can only be concluded that is surjective (see figure). Bijection A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence. A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follow. The function is bijective, if for all , there is a unique such that The