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Ok, so say I want to create a bunch of sets and functions for my to-be paper (that surely will get the attention of those comity members in Stockholm), and I want to identify them with the help of sub- and superscripts.

For example, take $I$ to be the set of all ice creams that exists, ever has existen and will exist in the world, and $me$ to stand for...well, for me. If I would like to denote the set of all ice creams that I have eaten during my life (and I wanted to do this with the help of either sub- or superscripts), what would be the correct or conventional way to write this; $I_{me}$ or $I^{me}$?

Further, what if I wanted to specify a function for how long time it would take for me to eat a certain set of ice creams (and I also wanted to do this with the help of either sub- or superscripts, instead of creating a general function, $f(X,Y)$ and then put myself in there), what would be the correct or conventional way to write this; $f_{me}(X)$ or $f^{me}(X)$?

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In this case, I'd go for subscripts. That's the way one generally (entering duck-and-cover mode here) distinguishes subsets satisfying certain criteria. See @axblount's answer for a more specific suggestion. –  Rick Decker Aug 31 '12 at 2:22
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You might consult the papers others have written about their ice cream and how long they've taken to eat it. I wouldn't depart from standard notation unless you have a good reason. But as long as you're clear about your notation, use whatever you're most comfortable with.

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Nicely said, +1 –  Rick Decker Aug 31 '12 at 2:18
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