March 8th, 2016, 11:14 AM  #1 
Newbie Joined: Mar 2016 From: Guatemala Posts: 4 Thanks: 0  Infinite sets
Hi. Could you help me with the following proof? Let X be an infinite set and Y an infinite set. Show that there exists an injective function f: X >Y and a surjective function g: Y > X Thanks 
March 8th, 2016, 11:26 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
Are you sure there is no more detail given about the two sets? I don't think the statement is true without some qualification. For example, if $X=\mathbb R$ and $Y=\mathbb N$.

March 8th, 2016, 11:32 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
If the only condition on X and Y are that they be infinite, the problem is "symmetrical" and it seems strange that we would be asked to prove that there is an "injective" function one way and a "surjective" function the other way.

March 8th, 2016, 11:32 AM  #4 
Newbie Joined: Mar 2016 From: Guatemala Posts: 4 Thanks: 0 
I know that X and Y are elementos of the natural Numbers. Other than that I don't have more details.

March 8th, 2016, 11:34 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
So we have $X \subset \mathbb N$ and $Y \subset \mathbb N$, with both $X$ and $Y$ being infinite? That makes more sense.

March 8th, 2016, 11:36 AM  #6 
Newbie Joined: Mar 2016 From: Guatemala Posts: 4 Thanks: 0 
Yes, both are in the natural numbers. And I'm sorry, I think I made a typo. X is a finite set 

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