Set Theory Problem Involving Partitions

[hmb]

This problem is from Hrbacek and Jech, Introduction to Set Theory, Third Edition, right at the end of chapter 2.

Homework Statement

Let $$A \neq$$ {}; let $$Pt(A)$$ be the set of all partitions of $$A$$. Define a relation $$\leq$$ in $$Pt(A)$$ by

$$S_{1} \leq S_{2}$$ if and only if for every $$C \in S_{1}$$ there is $$D \in S_{2}$$ such that $$C \subseteq D$$.

(We say that the partition $$S_{1}$$ is a refinement of the partition $$S_{2}$$ if $$S_{1} \leq S_{2}$$ holds.)

(a) Show that $$\leq$$ is an ordering.

DONE.

(b) Let $$S_{1}, S_{2} \in Pt(A)$$. Show that $$\{S_{1}, S_{2}\}$$ has an infimum. [Hint: Define $$S = \{C \cap D | C \in S_{1} and D \in S_{2}\}$$.] How is the equivalence relation $$E_{S}$$ related to the equivalences $$E_{S1}$$ and $$E_{S2}$$?

DONE; $$E_{S} = E_{S1} \cap E_{S2}$$

(c) Let $$T \subseteq Pt(A)$$. Show that inf$$T$$ exists.

(d) Let $$T \subseteq Pt(A)$$. Show that sup$$T$$ exists. [Hint: Let $$T'$$ be the set of all partitions $$S$$ with the property that every partition from $$T$$ is a refinement of $$S$$. Show that sup$$T'$$ = inf$$T$$.]

Homework Equations

$$a \in Pt(A)$$ is an upper bound of $$T$$ in the ordered set $$(Pt(A), \leq)$$ if $$x \leq a$$ for all $$x \in T$$.

$$a \in Pt(A)$$ is called a supremum of $$T$$ in $$(Pt(A), \leq)$$ if it is the least element of the set of all upper bounds of $$T$$ in $$(Pt(A), \leq)$$.

$$a \in Pt(A)$$ is a lower bound of $$T$$ in the ordered set $$(Pt(A), \leq)$$ if $$a \leq x$$ for all $$x \in T$$.

$$a \in Pt(A)$$ is called an infimum of $$T$$ in $$(Pt(A), \leq)$$ if it is the greatest element of the set of all lower bounds of $$T$$ in $$(Pt(A), \leq)$$.

The Attempt at a Solution

I have been trying to answer part (c). I figured I would need to generalise the method for proving part (b) but I cannot figure out how to do it. Then I thought maybe the hint for part (d) might be relevant to solving part (c), but I can't get my head around how that would work either.

 

[ystael]

Given any $$a\in A$$, what properties should the cell of $$a$$ in $$\inf T$$ have? How does that tell you how to compute the cell of $$a$$ in $$\inf T$$ from the members of $$T$$?

 

[hmb]

Is the cell of $$a$$ in $$infT$$ the equivalence class of $$a$$ modulo $$E_{infT}$$, i.e., $$[a]_{E_{infT}}$$? If so, then I think the properties the cell of $$a$$ in $$infT$$ would have would be as follows:

For all $$a \in A$$, for all $$x \in T$$, $$[a]_{E_{infT}} \subseteq [a]_{E_{x}}$$ and for all $$y \in Pt(A)$$, if $$[a]_{E_{y}} \subseteq [a]_{E_{x}}$$ then $$[a]_{E_{y}} \subseteq [a]_{E_{infT}}$$

Is that right?

 

[hmb]

OK, I think I have worked out part (c). I couldn't be bothered doing all the latex, so if you are interested I have attached my work as .JPG files.

 

[hmb]

. . . and here is the last page.

 

[hmb]

Sorry, the 5th-last line on the last page should read:

For every $$x \in B$$ and $$a \in A$$, $$\left[a\right]_{E_{x}} \subseteq \bigcap \left\{\left[a\right]_{E_{y}} \left| y \in T\right\}$$.

 

[hmb]

OK, I think I've got part (d) now as well, although I think the 'hint' was supposed to read: [Hint: Let $$T'$$ be the set of all partitions $$S$$ with the property that every partition from $$T$$ is a refinement of $$S$$. Show that sup$$T$$ = inf$$T'$$.] I have attached the proof.

 

[hmb]

Sorry, another mistake; in the first line it should say $$x \leq S$$, not $$x \in S$$.