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Is there any way to interpret the set of all functions from a set $X$ to a set $Y$?

There is an interpretation of it as the cartesian product of $X$-many copies of $Y$, but I am asking for a more fun, if you want, interpretation. Maybe something of combinatorial flavour?

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I had a professor who once was telling our class about an idea he had as an undergrad. Since the powerset of $X$ is just all $f:X\to\{0,1\}$ he considered the set of all $f:X\to[0,1]$. They're actually called Fuzzy sets, en.wikipedia.org/wiki/Fuzzy_set, I've never read up on them, but he warned us that he did not like the theory. –  doppz Mar 12 at 17:36

1 Answer 1

If you accept well-ordering, then consider X as a well-ordered set and then the functions from X to Y represent sequences of the elements of Y according to the well-ordering of X.

So, if X is the ordinal numbers then the functions represent sequences of elements of Y extending through the ordinals.

At a more mundane level, if X is specifically the positive integers, then the functions are what we would normally understand as infinite sequences of elements of Y.

Well, that's one interpretation.

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