Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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
1  
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
add comment

3 Answers

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.

share|improve this answer
add comment

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

share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.