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Partition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set." For example, one possible partition of $(1, 2, 3, 4, 5, 6 )$ is $(1, 3), (2), (4, 5, 6).$ Rudin, while defining integral on page $120$ starts like this,

Definition Let $[a, b]$ be a given interval. By a partition $P$ of $[a,b]$ we mean a finite set of points $x_0, x_1,..., x_n$, where $a=x_0\leq x_1\leq...\leq x_n=b$.

if all the points from $a$ to $b$ are in partition $P$ then where is the other partition, is it $\phi$?, not mentioned in the book.

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The confusion arises because of the shortage of words in the English language, and the contemporary emphasis on recycling and reusing. – copper.hat Jul 31 '12 at 6:46
Wikipedia: Partition of a set and Partition of an interval. The word has also other meanings in mathematics, see again Wikipedia. – Martin Sleziak Jul 31 '12 at 7:40
up vote 5 down vote accepted

This is a slightly different kind of partition. Here the idea is that the interval $[a, b]$ is being partitioned into sub-intervals $[x_0, x_1], [x_1, x_2], \ldots$.

As with the kind of partition you defined, the sub-intervals here completely cover the original set $[a, b]$. Unlike with the kind of partition you defined, the sub-intervals here are not exactly disjoint. Instead they are almost disjoint, since they overlap only at their endpoints.

Rudin says that the points $x_0, x_1,\ldots$ "are" the partition, but that is just because once you know those points, you know everything there is to know about the way that $[a,b]$ has been divided into sub-intervals. In a more general setting, with the definition you quoted, that is not the case.

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You're looking at two different definitions. The second definition is used to develop the theory of integration.

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The first definition of a partition is the one that is more generally used.

However, if the context of Rudin's book, he is likely trying to define the integral. This definition different. However, note that $[x_0, x_1]$, $(x_1, x_2]$, ..., $(x_{n-1}, x_n]$ is a partition in the first sense. However, Rudin's definition of partition does not account for all possible partition of $[a,b]$ in the first sense.

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