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Can someone please help me clarify the notations/definitions below:

Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s?

Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector consisting of any number between $0$ and $1$ inclusive (exclusive)?

As a related question, is there a reference web page for all such definitions/notations? Or do we just need to take note of them individually as we learn. Thanks.

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    The interpretations you made are the first ones I would think of. But there are too few symbols, too much mathematics. If the writer intends something else, (s)he would have said so. Even if the "natural" interpretation is the intended one, it is useful to remind the reader.2012-03-18
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    I only know the first one. $\{ 0, 1 \}$ is the binary set, sometimes denoted in as $\mathbb{B}.$ So, yes, $\{ 0, 1\}^{n} = \mathbb{B}^{n};$ understood as $n$-vectors in $\mathbb{B}.$2012-03-18
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    It might help to know what a cartesian product is ...2012-03-18

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The notation $\{0,1\}^n$ refers to the space of all $n$-length vectors consisting of $0$s and $1$s, while the notation $[0,1]^n$ ($(0,1)^n$) refers to the space of all $n$-length vectors consisting of real numbers between $0$ and $1$ inclusive (exclusive).

Edit: I often find wikipedia's list of mathematical symbols useful for looking up the meaning of symbols, although I'm not sure it would help with this question.

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    +1, but maybe it would be better to call $\left\{0,1{}\right\}^n$ a *set of (ordered) n-tuples* consisting of 0s and 1s. It certainly can be a space, because it has several structures imposed on it, e.g. order, Hamming distance, etc. But the "n-tuple" wording is slightly more general, I think...2012-03-18
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    @user825089 I have never seen tuple used to denote an set of elements without order. Could you provide an example?2012-03-18
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    I haven't either, but apparently the term exists as a natural extension of unordered pair: http://en.wikipedia.org/wiki/Unordered_pair (last sentence).2012-03-18
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The idea is simple… The abstract set of Topology

$$\displaystyle{\prod_{i\in I}X_i=\left\{x:I\rightarrow\cup_{i\in I}X_i \vert x(i)=x_i\in X_i,\;\forall i\in I \right\}}$$

where $x$ are continuous functions. (Also, you can see the continuous function as equivalence relation with $n$-length vector)

Examples:

  1. If $X_i=\{0,1\}, \forall i\in I$ then $$\displaystyle{\prod_{i\in I}X_i=\{0,1\}^I}$$ $$x=(x_i)_{i\in I}\in\{0,1\}^I$$

  2. If $X_i=[0,1], \forall i\in I$ then $$\displaystyle{\prod_{i\in I}X_i=[0,1]^I}$$

  3. Important case is the Cantor's set.

P.D.: Excuse my English, please.

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    Not that "simple" for a simple idea. What you're saying is right, but you're... a little intense for OP. I think. =P2012-03-18