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
a)2^10=1024
b)59049
c)1048576
d)9765625
There are n^10 functions from a set of ten elements to a set with n elements.
HOW THIS FORMULA ARIVES?
To find the number of different functions from a set with 10 elements to sets with different numbers of elements, we need to use the concept of cardinality and combinatorics. The number of functions from a set with m elements to a set with n elements is given by the formula:
n^m
where "^" denotes exponentiation. Applying this formula to each case, we can find the number of different functions for each scenario:
a) For a set with 2 elements, we have 2^10 = 1024 different functions.
b) For a set with 3 elements, we have 3^10 ≈ 59,049 different functions.
c) For a set with 4 elements, we have 4^10 ≈ 1,048,576 different functions.
d) For a set with 5 elements, we have 5^10 ≈ 9,765,625 different functions.
Therefore, the number of different functions from a set with 10 elements to sets with 2, 3, 4, and 5 elements are 1024, 59,049, 1,048,576, and 9,765,625, respectively.