# Question on a surjective map

• Jul 4th 2011, 05:03 AM
worc3247
Question on a surjective map
If $f:A \rightarrow B$ is surjective and A is finite, then B is finite. True or false?
I think this is false but my book says it is true.

I think its false by counter-example:
Let A= $(-\frac{\pi}{2},\frac{\pi}{2})$ and let f(x) = tan(x). Then B is infinite?

• Jul 4th 2011, 05:38 AM
emakarov
Re: Question on a surjective map
But your A is infinite as well despite the assumption.
• Jul 5th 2011, 07:53 AM
MoeBlee
Re: Question on a surjective map
Quote:

Originally Posted by worc3247
If $f:A \rightarrow B$ is surjective and A is finite, then B is finite. True or false?

Easy to prove:

Since A is finite, let h be a finite enumeration of A.

Let f be a surjection from A onto B.

Define a function g from B as follows:

g(y) = the least n such that f(h(n)) = y.

So g is an injection from B into a finite set.