A function
[katex]f:A\rightarrow B[katex]
is called injective function (or one one function) if
[katex](\forall x,y\in A)[f(x)=f(y)\Rightarrow x=y][katex]
if and only if
[katex](\forall x,y\in A)[x\neq y\Rightarrow f(x)\neq f(y)][katex]
if and only if
[katex](\forall X,Y\subseteq A)[X\subseteq Y\Rightarrow f(Y\setminus X)=f(Y)\setminus f(Y)][katex]
if and only if
[katex](\forall X,Y\subseteq A)[X\cap Y=\emptyset \Rightarrow f(X)\cap f(Y)=\emptyset][katex]
if and only if
[katex](\forall X,Y\subseteq A)[f(X\cap Y)=f(X)\cap f(Y)[katex]
if and only if
[katex](\forall X\subseteq A)[f^{-1}(f(X))=X, ~~~~f^{-1}(Y)=\{x\in A|f(x)\in Y\}~{\text{for~any }}~Y\subseteq B[katex]if and only if
[katex](\exists g:B\rightarrow A)g\circ f=id_A~~\text{where}~id_A~\text{is~identity~mapping~on}~A[katex]The proof that they are equivalent are left as exercise!!!