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Functions are usually (though not always) defined as a relation, which is a set of ordered pairs. In this sense, functions are merely sets. However, some ''functions" can only be described by proper classes, and not sets. A familiar example of this is the power set function $\mathcal{P}(X)$. Is there a name for these kinds of "functions", or are they just called functions as well, or is there perhaps not a standard terminology for them?

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    $\begingroup$ I've seen/heard "class function" used before. $\endgroup$ – Hayden Jan 7 '15 at 20:41
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Class function, as Hayden points in the comments, is the common term for describing a collection of ordered pairs which is not [necessarily] a set, and if $\langle x,y\rangle$ and $\langle x,z\rangle$ are both in this collection, then $y=z$.

In some contexts we just call them functions and expect the reader/listener/world-class violinist to infer from context whether or not the function is a set or a class (although in those contexts it's often unimportant, which is why we neglect to mention that information).

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