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November 15th, 2017, 05:22 AM   #1
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Counting Functions

Hi: I'm trying to understand counting functions between two sets.

For example, between two 3-element sets, there are 3^3=27 functions.

And there are 3!/0!=6 one-to-one functions.

What are the other 21 functions?

Thayer
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November 15th, 2017, 05:30 AM   #2
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Quote:
Originally Posted by thayer View Post
Hi: I'm trying to understand counting functions between two sets.

For example, between two 3-element sets, there are 3^3=27 functions.

And there are 3!/0!=6 one-to-one functions.

What are the other 21 functions?

Thayer
If a, b, c constitutes one set, and x, y, z constitutes the other set, you can list all 27 possible distinct functions yourself. I suggest you do so.

(1) a to x, b to x, c to x;

and so on.
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November 16th, 2017, 04:11 AM   #3
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With, as JeffM suggested, {a, b, c} as domain and {x, y, z} as range the 6 "one-to-one" 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 "one-to-one". 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 "one-to-one" cannot be "onto".

One example is a-> x, b-> x, c-> x. Another is a->x, b->y, c->y.
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November 17th, 2017, 07:13 AM   #4
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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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