$\{0,1\}^n$ and $[0,1]^n$ notations
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.
$\endgroup$ 32 Answers
$\begingroup$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.
$\endgroup$ 3 $\begingroup$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:
If $X_i=\{0,1\}, \forall i\in I$ then $$\displaystyle{\prod_{i\in I}X_i=\{0,1\}^I}$$ $$x=(x_i)_{i\in I}\in\{0,1\}^I$$
If $X_i=[0,1], \forall i\in I$ then $$\displaystyle{\prod_{i\in I}X_i=[0,1]^I}$$
Important case is the Cantor's set.
P.D.: Excuse my English, please.
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