There exists a set S such that the number of function f : S ?{1,2,3} is 1000

Question

There exists a set S such that the number of function f : S →{1,2,3} is 1000

true or false, please provide with explanation.

Answer 1

The statement is false.

 

A function f : S -> {1,2,3} is completely determined by its images f(s), where s ranges over S. For each s in S, there are 3 possible values to assign f(s). So the number of ways of defining f is 3^n if S is finite of cardinality n, and infinite if S is itself infinite. 

 

If there exists a set S for which the number of functions f : S -> {1,2,3} is 1000, then S is finite and the number of functions is a power of 3 (by argument of the previous paragraph). Since 1000 is not a power of 3, our original assumption was false. Hence, no such S exists.