A,B nonempty finite sets f is a bijection from A to B. Prove if $g:A\rightarrow B$ is injective then it's surjective. Is this true for infinite sets?
If f is a bijection from A to B all we can say is that A and B have the same cardinality because bijection means every element in A has a unique element in B and all elements in both are mapped.
If g is injective then that means every element in A has a unique element in B that it's mapped to, but because A and B are the same cardinality that means that every element in B must have an element it's mapped to in A, therefore it is surjective.
Let me know if any of the above proof is incorrect.
I don't know if this holds true for infinite sets though. I want to say no? Because if they're both infinite sets does that mean they have the same cardinality?