Simple Cardinal Exponentiation Example

Question

In general, for two sets $A$ and $B$, $|A^B|$=the number of functions from $B$ into $A$.

Can someone please show that $|\{1,2\}^{\{3,4,5\}}|=8$ by showing that there are eight functions from $\{3,4,5\}$ to $\{1,2\}$?

Answer 1

$(3,4,5) \to (1,1,1)$

$(3,4,5) \to (2,1,1)$

$(3,4,5) \to (1,2,1)$

$(3,4,5) \to (1,1,2)$

$(3,4,5) \to (2,2,1)$

$(3,4,5) \to (2,1,2)$

$(3,4,5) \to (1,2,2)$

$(3,4,5) \to (2,2,2)$

Answer 2

A function sends one element of the domain (that is 3, 4, and 5) to exactly one element in the co-domain. Therefore, there are two choices for each element in the domain: they can be either sent to 1 or sent to 2. Therefore, there are $2^3 = 8$ choices in total.