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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.

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

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  • $\begingroup$ I've seen this as well.+1 $\endgroup$ – Andres Mejia Jul 20 '17 at 20:00
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$B^A$ 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 $B$.

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

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  • $\begingroup$ +1. I think this is true in some other "Categories", maybe maps ( I think homeomorphisms) between rings, etc. $\endgroup$ – gary Jul 20 '17 at 20:02
  • $\begingroup$ @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.) $\endgroup$ – Andres Mejia Jul 20 '17 at 20:03
  • $\begingroup$ en.wikipedia.org/wiki/Homeomorphism so he may not have ... $\endgroup$ – user451844 Jul 20 '17 at 20:25
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    $\begingroup$ @RoddyMacPhee that is for topological spaces, not rings. Also, in Top., the morphisms would be continuous maps $\endgroup$ – Andres Mejia Jul 20 '17 at 20:29
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The name is exponentiation. See e.g. Smith, Romanowska - Post-Modern Algebra

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