# Name for the set of possible functions $A\to B$

Everybody knows the Cartesian product $A \times B$, where $|A\times B| = |A| \cdot |B|$.

But is there a name for the set of possible functions $A \to B$, where $|A \to B| = |B|^{|A|}$

E.g.

$$A = \{0, 1\} \\ f_1,f_2,f_3,f_4\colon A \to A \\ f_1\colon x \mapsto 0 \\ f_2\colon x \mapsto 1 \\ f_3\colon x \mapsto x \\ f_4\colon 0 \mapsto 1 \\ f_4\colon 1 \mapsto 0 \\$$

I know of the symmetric group, which is a similiar notion, but it only covers bijections, and not all possible functions.

• Often it's just $B^A$. – Ethan Bolker Jul 20 '17 at 19:58
• Simply write $B^A$. – Hagen von Eitzen Jul 20 '17 at 19:58
• en.wikipedia.org/wiki/Function_(mathematics)#Function_spaces – Jack Jul 20 '17 at 20:07
• The symmetric group on $X$, instead, is sometimes indicated as $X!$, again because its cardinality is $\lvert X! \rvert=\lvert X \rvert !$ – trying Jul 20 '17 at 21:08
• Your premise in the first line is false. My wife doesn't know the Cartesian product $A \times B$ – Χpẘ Jul 21 '17 at 20:17

Another piece of notation for this is $B^A$. The reason is because the cardinality of this set is the same as $|B|^{|A|}$.

• I've seen this as well.+1 – Andres Mejia Jul 20 '17 at 20:00

$B^A$ is common in mathematics.

Computer scientists (or at least the undersigned computer scientist) will sometimes prefer to cut the notational crap and declare "$A\to B$" to be the name of the set of maps from $A$ to $B$.

• I think I've only seen that with enclosing braces... – Eric Towers Jul 20 '17 at 23:47

Some typical notation would be $\mathrm{Hom}(A,B)$.This is the collection of maps from $(A,B)$, and in $\mathrm{Set}$, these are just usual functions.

However, this may just be habit for me, I think Alfred Yerger's notation is far more standard for sets.

• +1. I think this is true in some other "Categories", maybe maps ( I think homeomorphisms) between rings, etc. – gary Jul 20 '17 at 20:02
• @gary Yes this is true, but I didn't want to add to confusion with the Hom functor... It is true that the notation is used in other categories (homomorphisms you meant btw.) – Andres Mejia Jul 20 '17 at 20:03
• en.wikipedia.org/wiki/Homeomorphism so he may not have ... – user451844 Jul 20 '17 at 20:25
• @RoddyMacPhee that is for topological spaces, not rings. Also, in Top., the morphisms would be continuous maps – Andres Mejia Jul 20 '17 at 20:29

The name is exponentiation. See e.g. Smith, Romanowska - Post-Modern Algebra