# Injective function

## Function that preserves distinctness / From Wikipedia, the free encyclopedia

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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; that is, *x*_{1} ≠ *x*_{2} implies *f*(*x*_{1}) ≠ *f*(*x*_{2}). (Equivalently, *f*(*x*_{1}) = *f*(*x*_{2}) implies *x*_{1} = *x*_{2} in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of *at most* one element of its domain.^{[2]} The term *one-to-one function* must not be confused with *one-to-one correspondence* that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an *injective homomorphism* is also called a *monomorphism*. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.^{[3]} This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function $f$ that is not injective is sometimes called many-to-one.^{[2]}