Notations of a indexed famliy and a set of mappings

Question

Suppose $X$ and $Y$ are two sets.

A mapping $f: X\to Y$ can be seen as a family of elements of $Y$ indexed by $X$, so I see $f$ is also usually written as $(Y_x)_{x \in X}$. Isn't it?

The set of all possible mappings from $X$ to $Y$, I see, has the notation $Y^X$. I wonder why it is like the power of some number to another?

How about the power set of $X$ written as $2^X$? I seem to have seen some explanation somewhere, but forget what it says.

Thanks and regards!

This might be helpful, so would this. – Asaf Karagila Feb 08 '12 at 14:29 — Asaf Karagila, Feb 08 '12 at 14:29

score 3 · Accepted Answer · answered Feb 08 '12 at 14:29

The notation $Y^X$, I think, is motivated by the fact that if $X$ has $n$ elements and $Y$ has $m$ elements, then the set of functions from $X$ to $Y$ will have $m^n$ elements.

The notation $2^X$ for the powerset of $X$ comes from the one to one correspondence between the powerset of $X$ and the set of functions from $X$ to $2 = \{0, 1\}$.

Notations of a indexed famliy and a set of mappings

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