$\{0,1\}^n$ and $[0,1]^n$ notations

Question

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.

Answer 1

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.

Answer 2

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.