- #1

- 15

- 0

*Introduction to Set Theory*, Third Edition, right at the end of chapter 2.

## Homework Statement

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

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

(We say that the partition [tex]S_{1}[/tex] is a

*refinement*of the partition [tex]S_{2}[/tex] if [tex]S_{1} \leq S_{2}[/tex] holds.)

(a) Show that [tex]\leq[/tex] is an ordering.

**DONE.**

(b) Let [tex]S_{1}, S_{2} \in Pt(A)[/tex]. Show that [tex]\{S_{1}, S_{2}\}[/tex] has an infimum. [

*Hint:*Define [tex]S = \{C \cap D | C \in S_{1} and D \in S_{2}\}[/tex].] How is the equivalence relation [tex]E_{S}[/tex] related to the equivalences [tex]E_{S1}[/tex] and [tex]E_{S2}[/tex]?

**DONE; [tex]E_{S} = E_{S1} \cap E_{S2}[/tex]**

(c) Let [tex]T \subseteq Pt(A)[/tex]. Show that inf[tex]T[/tex] exists.

(d) Let [tex]T \subseteq Pt(A)[/tex]. Show that sup[tex]T[/tex] exists. [

*Hint:*Let [tex]T'[/tex] be the set of all partitions [tex]S[/tex] with the property that every partition from [tex]T[/tex] is a refinement of [tex]S[/tex]. Show that sup[tex]T'[/tex] = inf[tex]T[/tex].]

## Homework Equations

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

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

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

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

## 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.