this is an elementary doubt ,i am not asking advance mathematics ,please dont close this i have been having this doubt,
many-to one functions are not injective ,but they can be onto i mean let us take the following example
let $f : N \times N \mapsto N$ be a many to one function defined by $f(a,b)=a.b$ but how can we prove that the function is onto ????
in general how can we prove the surjectiveness of many to one functions
suppose let us take the number $n$ to be the count of the cardinality of elements that function binds and sends to the other side,i mean the codomain in the above example the value of $n$ is $2$ (as function $f$ sends the 2 elements $a$ and $b$ to the another side )
so how can we prove the surjectiveness of the function, i mean the definition is bit manipulated ,
the new version of the definition seems to be
"for every $y \in $ Codomain $\exists (x_1,x_2....x_n) \in $ Domain such that $f(x_1,x_2....x_n)=y$ , which is a new surjective function notion
please suggest your valuable comments in proving the surjectiveness of many to one functions (i think that word surjectiveness dont exist as the editor is showing red line,but that words gits rightly)
i mean ,how to prove that a many to one function is onto
thanks a lot,to one and all