Thread: Help with injective, surjective proof

Please help me with this, I'm not sure I understand it and don't know where to start

Let A and B be finite sets with |A| = |B|. Suppose that f: A--->B. Prove that f is injective if and only if f is surjective

Originally Posted by Dax

Let A and B be finite sets with |A| = |B|. Suppose that f: A--->B. Prove that f is injective if and only if f is surjective

Suppose that is injective but not surjective.
Notation:

But is not surjective so .
So what is the contradiction?

is there an easier way to explain that, none of that looks familiar, i'm only a few weeks into the class and that doesn't make sense to me at all

Originally Posted by Dax

is there an easier way to explain that, none of that looks familiar, i'm only a few weeks into the class and that doesn't make sense to me at all

Look, this is not a trivial problem. It usually makes use of the pigeon-hole principle.
You are not going to find a trivial proof. Any proof is equivalent to proving a form of the pigeon-hole principle.

That said, try this.
Suppose that is injective.
is collection of pair-wise disjoint subsets of .
But by the nature if functions .
From that how can conclude that is surjective?