# Difference of 2 notations with powerset

Let $\mathbb{N}, \mathbb{V}$ two sets, $\mathcal{P}(\ldots)$ means the power set of a set.

$\mathcal{P}({\mathbb{N}})\rightarrow \mathbb{V}$ can be the type of a function mapping a part of $\mathbb{N}$ into $\mathbb{V}$. I think $\mathcal{P}({\mathbb{N}})\rightarrow \mathbb{V} = \mathcal{P}({\mathbb{N}} \rightarrow \mathbb{V})$ always holds. Could anyone tell me what is the difference between these 2 notations? Is $\mathcal{P}({\mathbb{N}})\rightarrow \mathbb{V}$ always more conventional than $\mathcal{P}({\mathbb{N}} \rightarrow \mathbb{V})$?

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What does the arrow indicates? (Having never seeing such notation before...) – Asaf Karagila Feb 23 '13 at 0:18

Assuming that $\Bbb{N\to V}$ indicates the set of functions from $\Bbb N$ to $\Bbb V$, then the sets are not even the same.
1. In $\cal P(\Bbb{N)\to V}$ we have functions mapping subsets of $\Bbb N$ to $\Bbb V$.
2. In $\cal P(\Bbb{N\to V})$ we have sets of functions from $\Bbb N$ to $\Bbb V$.