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February 26th, 2018, 10:49 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory  Finite mapped on infinite set
Can a finite set ever be mapped onto an infinite set?

February 26th, 2018, 11:08 PM  #2 
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193 
No.

February 27th, 2018, 01:29 AM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory 
Couldn't, e.g., each member of a finite set be mapped countless times into an infinite set?

February 27th, 2018, 01:34 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond 
Then, contrary to the definition of a function, f(n) would have more than one value. How could you avoid that? Can you be more explicit as to exactly how each member of N would be mapped to S?

February 27th, 2018, 02:42 AM  #5 
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193 
In set theory, we have the notion of a relation and of a function. A function is a special relation, but most of the functionlanguage makes sense for relations really. So you mean, to take the finite set {*} and the infinite set N={0,1,2,3,...}, we want to define *R0, *R1, *R2, etc (meaning that * is related to 0, * is related to 1, etc.) This is a perfectly fine relation. A function, however, is a relation R such that for each x, there is EXACTLY one y such that xRy. If we have *R0 and *R1, this condition is violated since there are multiple numbers y such that *Ry. Last edited by skipjack; February 27th, 2018 at 02:55 AM. 
February 27th, 2018, 03:19 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra  
February 27th, 2018, 05:55 AM  #7 
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193  No, exactly is the correct word. This IS the most common usage of the term function by mathematicians, regardless of what high school teachers say. If you want "at most", it is nowadays called a partial function, among other names. 
February 27th, 2018, 06:55 AM  #8 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,921 Thanks: 774 Math Focus: Wibbly wobbly timeywimey stuff. 
Consider f to be the finite set $\displaystyle f = \{1, 0, 1 \}$. Then define $\displaystyle p: f \to \mathbb Z$ be the map defined by p(1) = all negative integers, p(0) = 0, and p(1) = all positive integers. Isn't this a surjective function? Or do I need more coffee this morning? Dan 
February 27th, 2018, 10:41 AM  #9  
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193  Quote:
 
February 27th, 2018, 12:36 PM  #10  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory  Quote:
Microm@ss, is there a way around the strict definition of a function in this case, like topsquark's example? What if f={1} such that p=(1, 1, 1...)? Possibly countless elements from one, but can they all be the same? Please forgive my atrocious notation.  

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