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If I can can define a binary string with $n$ bits as a vector in the space $\{0,1\}^n$; how can I define a vector in $\mathbb{R}^n$ for the interval $[0,1]$? Can I just write $[0,1]^n$?

Gus Kenny
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  • An element of that would indeed be a vector whose components are all between $0$ and $1$, inclusive. – Hagen von Eitzen Oct 19 '19 at 10:39
  • If this (or any other) notation is unfamiliar to you, also include the definition the first time you include it in your document, since it may also be unfamiliar to some of your readers. – GEdgar Oct 19 '19 at 10:47

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Let $A$ be an arbitrary set. Then $A^n$ denotes the set $A\times A\times \dots \times A$, where $A$ appears $n$ times in the cartesian product. You can call $A^n$ a cartesian power.

Therefore $A^n$ is the set of $n$-tuples $(a_1,a_2,\dots,a_n)$ where each $a_i$ belongs to $A$.

This applies to any set $A$. So, $\Bbb R^n$ is the set of $n$-tuples of reals, and $[0,1]^n$ is the set of $n$-tuples of numbers between $0$ and $1$. And $\{0,1\}^n$ is the set of bit strings of length $n$.

Side note: to consider $\{0,1\}^n$ a vector space, you have first to define a structure of field on $\{0,1\}$. See GF(2). And $[0,1]^n$ is not a vector space, though it's a subset of the vector space $\Bbb R^n$.

Jean-Claude Arbaut
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  • Thanks, I thought that might be the case. And thanks for the clarification on terminology. That makes sense. – Gus Kenny Oct 19 '19 at 11:04