# How to denote a singleton?

I usually denote a set whose elements are distinct by $\{a_p\}_{p \in P}$. And I have a function $f$ which takes a set as argument, so we could write $f(\{a_p\}_{p \in P})$. My question is how to write it when $f$ takes a set which contains only 1 element (a singleton). If we write it $f(\{a_p\})$, it is still not obvious that the set is a singleton; if we write $f((a_p))$ or $f(a_p)$, it contradicts to the fact that f accepts only set as argument.

Does anyone have any idea?

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what's wrong with $\{ a \}$ and $f(\{ a \})$? – user20266 Dec 25 '11 at 14:06
Different elements of a set are distinct. Are you working with multisets? In a multiset elements can appear with repetition. Suppose you are just working with sets. Using your notation, if $p^{*} \in P$ then $f(\{ a_{p^{*}} \} )$ seems to work as you want. – Jay Dec 25 '11 at 14:55
The problem, as Marc says in his answer, is that you’re using a poor notation for sets. To me $f(\{a_p\})$ obviously does mean that the argument of $f$ is a singleton; if it weren’t, it would be written $f(\{a_p:p\in P\})$ or the like. – Brian M. Scott Dec 25 '11 at 19:16

The notation you use seems to me to be a recent invention (by analogy of the sequence $(a_i)_{i\in\mathbf N}$ or a matrix $(a_{i,j})_{i,j=1,\ldots,n}$); I don't recall ever having seen it more than a few years ago (but of course this ned not mean very much). If it is a recent invention, it is not a very fortunate one. I have always learned to denote sets as $\{a_p\mid p\in P\,\}$ or $\{a_p: p\in P\,\}$ and this avoids the confusion you mention. So $(f\{a_p\})_{p\in P}$ would be a sequence of values obtained by applying $f$ to singletons, quite distinct from $f(\{a_p\mid p\in P\,\})$.

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In the case that $A=\{a_p\}_{p\in P}$ is a singleton, then $f(A)$ ...
Thomas's suggestion of dropping the subscript and writing $\{a\}$ and $f(\{a\})$ is also good.