I am studying the procedure for bucket sort from Introduction To Algorithms by Cormen et al, which assumes that the input is generated by a random process that distributes the elements uniformly and independently over the interval $[0,1).$ What does this mean? Why there is no "]" closing bracket for the interval?

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    $\begingroup$ Irrelevant tags. $\endgroup$ – WacDonald's Aug 12 '12 at 16:44
  • $\begingroup$ @WacDonald's you can edit the tags if you think they are irrelevant . I encountered this while studying algorithms and since this algorithm assumes about probablility distribution of inputs , I put those tags . Since it has to do with sets , the elementary set theory is also justified in my opinion. $\endgroup$ – Geek Aug 12 '12 at 16:51
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    $\begingroup$ The notation is called half-closed interval. A closed interval $[0, 1]$ includes the end points $0, 1.$ An open interval $(0, 1)$ does not include the end points $0, 1.$ A half-closed interval is closed on one side, open on the other side. So $[0, 1)$ includes $0$ but does not include $1.$ $\endgroup$ – user2468 Aug 12 '12 at 16:56
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    $\begingroup$ @Geek: It is the usual notation for intervals. The interval $[a,b)$ is sometimes called half-open, as is $(a,b]$. The interval $(a,b)$ is an open interval, while $[a,b]$ is a closed interval. $\endgroup$ – André Nicolas Aug 12 '12 at 16:56
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    $\begingroup$ In Section 15.3 they define a closed interval along with notation, they mention open and half-open intervals but do not provide notation. $\endgroup$ – copper.hat Aug 12 '12 at 17:03

The notation $[0,1)$ refers to the set of all real numbers $x$ such that $0\le x\lt 1$. Another common notation for this set is $[0,1[$; which is more common often depends on the language in which the author was educated.

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    $\begingroup$ Interval notation? $\endgroup$ – GEdgar Aug 12 '12 at 17:02
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    $\begingroup$ I never saw the second notaion... $\endgroup$ – Belgi Aug 12 '12 at 19:16
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    $\begingroup$ It is less common, and I have not seen it in a North-American-style calculus book. But one does see it more frequently nowadays. For me it is visually a little harder to separate from the square bracket running the "normal" way. But is probably just a matter of familiarity. $\endgroup$ – André Nicolas Aug 12 '12 at 19:23
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    $\begingroup$ I have tended to think of the second notation as the "French" notation, but I don't know how accurate that is. $\endgroup$ – Michael Hardy Aug 12 '12 at 23:54
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    $\begingroup$ I think the second notation was introduced by the Bourbaki in order to prevent confusion with ordered pairs. $\endgroup$ – Vincent Pfenninger Jul 2 '13 at 19:36

In general, there are four possible variants for what we call intervals. The parenthesis $($ and $)$ are related to the strict inequality $<$, while these ones $[$ and $]$ are related to the weaker $\leq$. So, when we want to denote intervals, we use them as follows

$$\{x \text{ such that } a<x<b\}=(a,b)$$

$$\{x \text{ such that } a\leq x<b\}=[a,b)$$

$$\{x \text{ such that } a<x \leq b\}=(a,b]$$

$$\{x \text{ such that } a \leq x \leq b\}=[a,b]$$

You might also see $]a,b[$ for $(a,b)$, that is, the reversed $]$ are used just like parenthesis.

There is also what we call "rays" (which are also intervals), which involve a "one sided" inequality:

$$\{x \text{ such that } a<x\}=(a,\infty)$$

$$\{x \text{ such that } a\leq x\}=[a,\infty)$$

$$\{x \text{ such that } x \leq b\}=(-\infty,b]$$

$$\{x \text{ such that } x < b\}=(-\infty,b)$$

and what we usually denote by the real line

$$\{x \text{ such that }x\in \Bbb R \}=(-\infty,\infty)$$


This means that your interval goes from 0 to 1 but 1 itself is not included in the interval. You're random number process will generate a number between 0 and 1 (1 not included). We call this a half closed interval. Sometimes they write in textbooks [0,1[ in stead of [0,1), that's the same.

Sorry if the explanation is not mathematical enough. I'm a computer scientist ;-).


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