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 infT exists.

(d) Let T \subseteq Pt(A). Show that supT 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 supT' = infT.]

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 <br /> T&#039;<br /> be the set of all partitions <br /> S<br /> with the property that every partition from <br /> T<br /> is a refinement of <br /> S<br />. Show that supT = infT&#039;.] I have attached the proof.

 

[hmb]

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