Is it true, if there is no surjection of A onto B, then there must be injection A onto B??

Question

A is any set. If there is no surjection of A onto B, then there must be injection A onto B.

Is it true?? if so give me the proof, please.

Answer 1

The answer to the question you asked is obviously no, because an injection of $A$ onto $B$ is surjective; that's what "onto" means. You meant to ask this:

If there is no surjection of $A$ onto $B$ must there exist an injection from $A$ into $B$?

The answer to that question is yes, assuming the Axiom of Choice: Let $M$ be the set of all ordered pairs $(F,f)$ such that $F\subset A$ and $f:F\to B$ is injective. Define a partial order on $M$ by saysing that $(F_1,f_1)\le(F_2,f_2)$ if $F_1\subset F_2$ and $f_2|_{F_1}=f_1$.

Zorn's Lemma shows that $M$ has a maximal element; now if $(F,f)$ is maximal the fact that there is no surjection from $A$ onto $B$ shows that $F=A$.