How many distinct functions can be defined from set A to B? In my discrete mathematics class our notes say that between set $A$ (having $6$ elements) and set $B$ (having $8$ elements), there are $8^6$ distinct functions that can be formed, in other words: $|B|^{|A|}$ distinct functions. But no explanation is offered and I can't seem to figure out why this is true. Can anyone elaborate?

The cardinality of $B^A$ is the same if $A$ (resp. $B$) is replaced with a set containing the same number of elements as  $A$ (resp. $B$).
Set $b = |B$|. When $b \lt 2$ there is little that needs to be addressed, so we assume $b \ge 2$. Assume $|A| = n$.
A well known result of elementary number theory states that if $a$ is a natural number and $0 \le a \lt b^n$ then it has one and only one base-$\text{b}$ representation,
$$\tag 1  a = \sum_{k=0}^{n-1} x_k\, b^k \text{ with } 0 \le x_k \lt b$$
Associate to every $a$ in the initial integer interval $[0, b^n)$ the set of ordered pairs  
$$\tag 2  \{(k,x_k) \, | \, 0 \le k \lt n \text{ and the base-}b \text{ representation of } a \text{ is given by (1)}\}$$
This association is a bijective enumeration of $[0, b^n)$ onto the set of all functions 
mapping $[0,n-1]$ to $[0,b-1]$.
Since $[0, b^n)$ has $b^n$ elements, we know how to count all the functions from one finite set into another.
