# Is this an equivalent form of Axiom of Choice? [duplicate]

It is known that Axiom of Choice implies the following statement:

For each two sets $A$ and $B$, there is a one to one function from $A$ to $B$ iff there is a function from $B$ onto $A$

Is above statement an equivalent form of Axiom of Choice or it is strictly weaker?

• Are you sure the above statement is correct? – Tomasz Kania Aug 21 '14 at 11:58
• Also relevant: math.stackexchange.com/questions/732051/… Also see this: mathoverflow.net/questions/135945/… – Asaf Karagila Aug 21 '14 at 12:08
• (Note that as written the statement is false, since if $A$ is empty and $B$ is not, there is an injection but there is no surjection; if we exclude the case where $A$ is empty, then one direction is trivial without the axiom of choice. Every injection from a non-empty set has a surjective inverse.) Also the axiom of choice easily implies this, since it is equivalent to a stronger fact, that we can take the two functions to be inverses. – Asaf Karagila Aug 21 '14 at 12:10