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thayer November 15th, 2017 05:22 AM

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

JeffM1 November 15th, 2017 05:30 AM

Quote:

Originally Posted by thayer (Post 584308)
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.

Country Boy November 16th, 2017 04:11 AM

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.

mrtwhs November 17th, 2017 07:13 AM

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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