How many different functions are there from a set with 10 elements to sets with the following numbers of elements?

a) 2
b) 3
c) 4
d) 5

To find the number of different functions from a set with 10 elements to sets with a given number of elements, we need to consider the cardinality (i.e., the number of elements) of the domain and range sets.

a) When the range set has 2 elements, each element in the domain set (of 10 elements) can be mapped to either of the 2 elements in the range set. Since there are 2 choices for each element in the domain set, the total number of different functions is given by 2^10 = 1024.

b) Similarly, when the range set has 3 elements, each element in the domain set can be mapped to any of the 3 elements in the range set. Thus, there are 3 choices for each element in the domain set, leading to a total of 3^10 = 59049 different functions.

c) Likewise, when the range set has 4 elements, each element in the domain set can be mapped to any of the 4 elements in the range set. Consequently, there are 4 choices for each element in the domain set, resulting in a total of 4^10 = 1048576 different functions.

d) Again, when the range set has 5 elements, each element in the domain set can be mapped to any of the 5 elements in the range set. Therefore, there are 5 choices for each element in the domain set, yielding a total of 5^10 = 9765625 different functions.

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