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 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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