How many distinct functions can be defined from set A to B

Question

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?

Answer 1

Let set $A$ have $a$ elements and set $B$ have $b$ elements. Each element in $A$ has $b$ choices to be mapped to. Each such choice gives you a unique function. Since each element has $b$ choices, the total number of functions from $A$ to $B$ is $$\underbrace{b \times b \times b \times \cdots b}_{a \text{ times}} = b^a$$

Answer 2

Let's say for concreteness that $A$ is the set $\{p,q,r,s,t,u\}$. Let's try to define a function $f:A\to B$.

What is $f(p)$? It could be any element of $B$, so we have 8 choices.

What is $f(q)$? It could be any element of $B$, so we have 8 choices.

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What is $f(u)$? It could be any element of $B$, so we have 8 choices.

So there are $8\cdot8\cdot8\cdot8\cdot8\cdot8 = 8^6$ ways to choose values for $f$, and each possible set of choices defines a different function $f$. So that's how many functions there are.