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What does the notation $\{ 1,2 \}^{\mathbb{N}}$ mean? I have to build a bijection $\{ 1,2 \}^{\mathbb{N}} \to \{ 3,4 \}^{\mathbb{P}} $ ($\mathbb{P}$ denotes the set of odd numbers) but have no idea what is the set $\{ 1,2 \}^{\mathbb{N}}$. Any hint?

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Generally $X^Y$ means all functions from $Y$ to $X$. In this case $\{1,2\}^{\mathbb N}$ means all sequences with values 1 or 2.

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  • $\begingroup$ sequences or functions? $\endgroup$ – Leox Dec 10 '14 at 9:43
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    $\begingroup$ @Leox whatever you prefer. There is no essential difference between a sequence with indexset $\mathbb N$ and a function with domain $\mathbb N$. Personally I would go for functions. $\endgroup$ – drhab Dec 10 '14 at 9:47
  • $\begingroup$ Thanks, I see. Then what sequence corresponds to a function of $\{ 3,4 \}^{\mathbb{P}}$? $\endgroup$ – Leox Dec 10 '14 at 9:56
  • $\begingroup$ @Leox Exactly the fact that this question arises is on its own a good reason to do it with functions in this context. In the answer above $\mathbb P$ is not mentioned, so there was no bothering about it. $\endgroup$ – drhab Dec 10 '14 at 10:02
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In general, if $A$ and $B$ are sets, then $A^B$ is the set of all functions from $B$ to $A$.

In the case when $B$ is the natural numbers, that means $A^\mathbb N$ is the set of all infinite sequences of elements of $A$.

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