# Notation: Set of all functions with finite number of variables from $A$ to $B$?

The setting here is basic set theory. We denote by $B^A$ the set of all functions from $A$ to $B$. Also, to describe a function with $n$ variables, such that the variables come from $A$ and the function takes values in $B$, we can use $A^n$, the set of all $n$-tuples taking values in $A$ (formally - The set of all functions from $\{1,2,\dots,n\}$ to $A$).

So $B^{A^n}$ is the set of all functions of artity $n$ from $A$ to $B$.

The question: what is the standard notation (or notations) for the set $\bigcup_{n=1}^\infty B^{A^n}$, the set of all functions of arity at least 1 from $A$ to $B$?

Arity of at least 0 is also good (maybe $B^{A^*}$ is used in this case?)

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I will use some shorthand notation for sure (yours is good). I just want to check if there isn't already some standard notation. – Gadi A Feb 13 '12 at 13:22
You may want to consider whether you really need that set. For many purposes it is easier to identify each univariate function with a bivariate function that is constant in the second variable, etc. – Niels Diepeveen Feb 13 '12 at 14:42

I don't think there is a standard notation for that. In any case, $\bigcup_{n=1}^\infty B^{A^n}$ is pretty clear but considering just using words and a simpler notation such as ${\cal F}(A,B)$.