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Question for Intermediate Analysis (Please Help!)

AutGuy98

New member
Sep 11, 2019
20
Hey guys,

Got what seems like a simple problem at first, but has been giving me difficulty in trying to prove. Please help me with it and show me what to do because I have no clue really. Here is what it asks:

Prove that if A is a countable set and the function f:A\impliesB is onto, then B is either countable or finite.

Thanks in advance to whomever helps me out with this. The assistance is greatly appreciated.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,490
For each $b\in B$ choose a single $a_b\in A$ such that $f(a_b)=b$. Then consider the restriction of $f$ to $A'=\{a_b\mid b\in B\}$. Show that this is a bijection from $A'$ to $B$. As a subset of $A$, the set $A'$ is either countable or finite, and so is $B$.