Final answer:
There are 262,144 different functions possible from set a to set b and 1,679,616 different functions from set b to set a, calculated by raising the number of elements in the target set to the power of the number of elements in the source set.
Step-by-step explanation:
The question at hand involves counting the number of different functions that can be formed from one set to another. Specifically, we have a set a with 6 elements and a set b with 8 elements. To find the number of different functions from a to b, we consider that each element in a can be mapped to any of the 8 elements in b.
Since there are 6 elements in a, and each element has 8 choices in b, we use the formula for the number of functions from one set to another: |b|^|a|, where |a| and |b| denote the number of elements in set a and set b respectively.
Therefore, the number of functions from a to b is 8^6. In the reverse case, for functions from b to a, each of the 8 elements in b has 6 choices in a. Hence, the number of functions is 6^8. To express this in numerical terms: there are 262,144 different functions possible from a to b, and 1,679,616 from b to a.