Prove: Partitions and refinements

Question

Problem: Let $ R $ be the set of partitions of a real interval. Then for all elements in $ R $, every pair of elements has an upper bound.

I am having trouble structuring the proof; and intuitively understanding why every pair of elements has an upper bound.

Answer 1

Let the interval be $[a,b]$. A partition $X$ of $[a,b]$ is a finite sequence $X=\langle x_0,x_1,\ldots,x_n\rangle$ of points of $[a,b]$ such that $x_0=a$, $x_n=b$, and $x_0<x_1<\ldots<x_n$. A partition $Y=\langle y_0,\ldots,y_m\rangle$ refines $X$ if and only if $\{x_0,\ldots,x_n\}\subseteq\{y_0,\ldots,y_m\}$: every partition point of $X$ is a partition point of $Y$ or, equivalently, each interval of $Y$ is contained in an interval of $X$.

For reflexivity, you need only show that each partition of the interval is a refinement of itself; that’s completely trivial.

For transitivity you must show that if $Y$ refines $X$, and $Z$ refines $Y$, then $Z$ refines $X$; that also is a very straightforward application of the definition of refines.

Suppose that $X=\langle x_0,\ldots,x_n\rangle$ and $Y=\langle y_0,\ldots,y_m\rangle$ are partitions of $[a,b]$; you want a partition $Z=\langle z_0,\ldots,z_r\rangle$ that refines both $X$ and $Y$ and has the further property that if $W$ refines $X$ and $Y$, then $W$ refines $Z$. In other words, you want $Z$ to be the coarsest possible common refinement of $X$ and $Y$.

• What partition points does $Z$ absolutely have to include in order to refine in order to refine both $X$ and $Y$?

• Does $Z$ have to include any partitions points beyond that bare minimum?

Answer 2

Since partitions are just finite sets of reals containing endpoints ,then consider a following situation.

Namely one partition $X$ is refinement of other partition $Y$ iff $Y \subseteq X$.

Then we have a situation that for relation $<$ as defined above ,following holds:

$$X<Y \Leftarrow \Rightarrow X \subseteq Y$$

The inclusion relation is both reflexive and transitive,thus so must be your defined relation.

As for upper bounds,if $X$ and $Y$ are partitions then their union is their upper bound,since union of finite sets is finite,and their union refines both of them.