So.. what exactly is Partition of an Interval

Question

I have been researching about Partition of an Interval, and I'm quite confused.

Some articles(Peoples) say Partition of $[a,b]$ is a finite sequence of $ a = x_0 < x_1 < \cdot\cdot\cdot < x_n=b$ which is $P={\{x_0, x_1, \cdot\cdot\cdot, x_n}\}$ and $[x_0, x_1], [x_1, x_2], \cdot\cdot\cdot, [x_{n-1}, x_n]$ is the subintervals of Partition P

while others say its a set of subdivided subintervals $P={\{[x_0, x_1], [x_1,x_2]}, \cdot\cdot\cdot, [x_{n-1}, x_n\}$

so as in the title, what exactly is a Partition of $[a,b]$

score 1 · Accepted Answer · answered Apr 05 '22 at 15:21

1

Sometimes the same term gets used in multiple different ways in mathematics. This can be extremely annoying (cf. the common confusion around the notion of "function," discussed e.g. here). Technically, this is one such occasion.

However, note that the two definitions cited are equivalent in a precise sense: there is an explicit way to translate between the two notions. Given a "first-type partition" $\langle a=x_0,x_1,...,x_n=b\rangle$, we get a corresponding "second-type partition" $\{[a=x_0, x_1], [x_1, x_2],...,[x_{n-1},x_n=b]\}$ (and we can translate in the other direction too). This means that neither notion has more, or different, information than the other, and the conflation of the two is relatively benign.

answered Apr 05 '22 at 15:21

Noah Schweber

245,398

so we can think of ${{a=x_0, x_1, \cdot\cdot\cdot, x_n=b}}$ as a partition "acting" an a interval $[a,b]$ and ${{[x_0, x_1], [x_1, x_2], \cdot\cdot\cdot, [x_{n-1}, x_n]}}$ as the result of that action, and vice versa? – Apr 05 '22 at 15:29
@JohnDoe Sounds right, although I'd avoid the term "acting" in this context since it has an established (and unrelated) meaning in mathematics. – Noah Schweber Apr 05 '22 at 15:30
what representation would you recommend others to use, in studies or in writing an article using partition – Apr 05 '22 at 17:18
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@JohnDoe I don't think there's a particular advantage to either, as long as the author is explicit and consistent in their usage. – Noah Schweber Apr 05 '22 at 17:20
Somehow this nice answer hasn't received any votes so far. +1 there. – Paramanand Singh Nov 06 '22 at 08:47

So.. what exactly is Partition of an Interval

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