Hom- functor for set theory

Question

So , I missed one class of set theory and my instructor taught us something related to Hom- functor that I could not understand or find on the internet . He just wrote three lines : $$ Hom(A,B)=\langle f\mid f \text{ is a function from } A \text{ to } B\rangle $$ $$|Hom(A,B)|=|B|^|A|$$ and $$|Hom(\emptyset , \emptyset )|=0^0=1.$$

Please explain what he wanted to tell.

Answer 1

Sets $A$ and $B$ induce the set of functions $f:A\to B$.

This set of functions can be denoted as $\mathsf{Hom}(A,B)$.

(Btw, another notation for it is $B^A$)

The cardinality of $\mathsf{Hom}(A,B)$ equals $|B|^{|A|}$.

(For instance wonder how many functions $A\to B$ exist with $A=\{1,2,3\}$ and $B=\{6,9\}$. The answer to that is $|B|^{|A|}=2^3=8$)

There is exactly one function $\varnothing\to\varnothing$ which is the empty function.

This is consistent with $|\mathsf{Hom}(\varnothing,\varnothing)|=|\varnothing|^{|\varnothing|}=0^0=1$.

Answer 2

The first line is saying that $\text{Hom}(A,B)$ is the set of all functions from $A$ to $B$. (In some contexts, it's restricted to linear functions.)

The second line is saying that the number of such functions is the number of elements in $B$, raised to the power of the number of elements in $A$.

The third line is saying that there is only one function from the empty set to itself. (This is the "empty function".)