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 February 26th, 2018, 09:49 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 410 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, 10:08 PM #2 Senior Member   Joined: Oct 2009 Posts: 783 Thanks: 280 No.
 February 27th, 2018, 12:29 AM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 410 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, 12:34 AM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,935 Thanks: 1129 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, 01:42 AM #5 Senior Member   Joined: Oct 2009 Posts: 783 Thanks: 280 In set theory, we have the notion of a relation and of a function. A function is a special relation, but most of the function-language 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 01:55 AM.
February 27th, 2018, 02:19 AM   #6
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Quote:
 Originally Posted by Micrm@ss A function, however, is a relation R such that for each x, there is EXACTLY one y such that xRy.
"At most" rather than "exactly".

February 27th, 2018, 04:55 AM   #7
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Quote:
 Originally Posted by v8archie "At most" rather than "exactly".
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, 05:55 AM #8 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,159 Thanks: 878 Math Focus: Wibbly wobbly timey-wimey 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, 09:41 AM   #9
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Quote:
 Originally Posted by topsquark 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
But then, for example, p(1) equals both 1 and 2, which is not allowed.

February 27th, 2018, 11:36 AM   #10
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Quote:
 Originally Posted by topsquark 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
topsquark seems to have defined most closely to what I was getting at -- say p(-1) representing all negative integers, perhaps a surjective function. I think my main concern is whether one may repeat beyond limit identical elements of the domain (below). Would "exactly" allow that?

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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