
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 26th, 2018, 09:49 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 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: 439 Thanks: 147 
No.

February 27th, 2018, 12:29 AM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 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: 7,852 Thanks: 1077 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: 439 Thanks: 147 
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 01:55 AM. 
February 27th, 2018, 02:19 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,355 Thanks: 2469 Math Focus: Mainly analysis and algebra  
February 27th, 2018, 04:55 AM  #7 
Senior Member Joined: Oct 2009 Posts: 439 Thanks: 147  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: 1,879 Thanks: 761 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, 09:41 AM  #9  
Senior Member Joined: Oct 2009 Posts: 439 Thanks: 147  Quote:
 
February 27th, 2018, 11:36 AM  #10  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 374 Thanks: 26 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.  

Tags 
finite, infinite, mapped, set 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Infinite field with finite characteristic  IAmABread  Abstract Algebra  13  June 8th, 2017 12:27 PM 
Finite and Infinite Sets  Luiz  Real Analysis  5  April 22nd, 2015 10:15 AM 
Finite and Infinite Sets  Luiz  Real Analysis  4  April 8th, 2015 05:56 AM 
Finite and Infinite Sets  Luiz  Real Analysis  1  April 4th, 2015 11:58 AM 
finite or infinite?  mathdigger  Number Theory  5  September 3rd, 2010 09:37 PM 