November 15th, 2017, 06:22 AM  #1 
Newbie Joined: Nov 2017 From: Hooksett NH Posts: 1 Thanks: 0  Counting Functions
Hi: I'm trying to understand counting functions between two sets. For example, between two 3element sets, there are 3^3=27 functions. And there are 3!/0!=6 onetoone functions. What are the other 21 functions? Thayer 
November 15th, 2017, 06:30 AM  #2  
Senior Member Joined: May 2016 From: USA Posts: 1,256 Thanks: 523  Quote:
(1) a to x, b to x, c to x; and so on.  
November 16th, 2017, 05:11 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 
With, as JeffM suggested, {a, b, c} as domain and {x, y, z} as range the 6 "onetoone" functions are a> x, b> y, c> z a> x, b> z, c> y a> y, b> x, c> z a> y, b> z, c> x a> z, b> x, c> y a> z, b> y, c> x The other 21 functions are, of course, those that are NOT "onetoone". That is, more than one of the members of the domain are mapped to the same member of the range. Of course since the domain and range have the same finite cardinality a function that is not "onetoone" cannot be "onto". One example is a> x, b> x, c> x. Another is a>x, b>y, c>y. 
November 17th, 2017, 08:13 AM  #4 
Senior Member Joined: Feb 2010 Posts: 701 Thanks: 136 
I think there are only 18 functions from {a,b,c} to {x,y,z}. The total of 27 would include things like: a > x, a > y, a > z which is a relation but not a function. 

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