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Counting FunctionsHi: 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 |

Quote:
(1) a to x, b to x, c to x; and so on. |

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. |

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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