A function $latex f:A \rightarrow B$ is called injective function (or one one function) if for any $latex x,y\in A$ it follows that $latex f(x)=f(y)$ implies $latex x=y$.

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 $latex f:A \rightarrow B$ is called injective function (or one one function) if for any $latex x,y\in A$ it follows that $latex x\neq y$ implies $latex f(x)\neq f(y)$.

The second one is:

A function $latex f:A \rightarrow B$ is called injective function (or one one function) if for any $latex X,Y\subseteq A$ with $latex X\subseteq Y$ it follows that $latex f(Y\setminus X)=f(Y)\setminus f(X)$.

The third one is:

A function $latex f:A \rightarrow B$ is called injective function (or one one function) if for any $latex X,Y \subseteq A$, if $latex X\cap Y=\emptyset$, then $latex f(x)\cap f(Y)=\emptyset$.

The fourth one is:

A function $latex f:A \rightarrow B$ is called injective function (or one one function) if for any $latex X, Y \subseteq A$, it follows that $latex f(X\cap Y)=f(X)\cap f(Y)$.

The fifth one is:

A function $latex f:A \rightarrow B$ is called injective function (or one one function) if for any $latex X\subseteq A$, it follows that $latex f^{-1} (f(X))=X$ with $latex f^{-1}(Y)=\{x\in A|f(x)\in Y\}$ for any $latex Y\subseteq B$.

The sixth one is:

A function $latex f:A \rightarrow B$ is called injective function (or one one function) if there exists a function $latex g:B\rightarrow A$, such that $latex g\circ f=id_A$, with $latex id_A$ is the identity function on $latex A$.

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