A function f:A –> B is called injective function (or one one function) if for any x and y in A it follows that f(x)=f(y) implies x=y.

Other than the above definition, there are several equivalent definitions. Here they are.

The first one is of course the contraposition of the previous one.

A function f:A–> B is called injective function (or one one function) if for any x and y in A it follows that x≠y implies f(x)≠f(y).

The second one is:

A function f:A–> B is called injective function (or one one function) if for any X,Y⊆A with latex X⊆Y it follows that f(Y\X)=f(Y)\ f(X).

The third one is:

A function f:A–> B is called injective function (or one one function) if for any X,Y ⊆ A, if X∩ Y=Ø, then f(x)∩f(Y)=Ø.

The fourth one is:

A function f:A–> B is called injective function (or one one function) if for any X, Y ⊆ A, it follows that f(X ∩ Y)=f(X) ∩ f(Y).

The fifth one is:

A function f:A–> B is called injective function (or one one function) if for any X⊆A, it follows that f^{-1}(f(X))=X with f^{-1}(Y)={x∈A|f(x)∈ Y} for any Y ⊆ B.

The sixth one is:

A function f:A–> B is called injective function (or one one function) if there exists a function g:B–>A, such that g• f=id_A, with id_A is the identity function on A.

The proof that they are equivalent are left as exercise!!!