I am just wondering if counting number of possible functions such as $f\circ f$ ($3$ sets of numbers) is the same process as counting function from sample set $A$ to $B$ ($2$ sets only). Thanks!
2026-03-26 01:26:50.1774488410
Is counting how many composition functions are there same as counting other functions?
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Your question is a little big ambiguous so I’m going to try and interpret it for you. I think you are asking this “Is the method to count the number of possible functions $ A \to B $ the same as the method to count all the possible functions that arrise from composing two functions together”.
In more formal terms if you have sets $A$, $B$ and $C$ is $|C^A|$ (that is the cardinality of the set of all functions from A to C) the same as $|C^B \circ B^A|$?
The answer is no. You can show this by considering the case where $B$ has only one element. In this case there can only be $|C|$ diffrent functions as a result of this composition.