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Set Theory Problem Involving Partitions

  • Thread starter hmb
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  • #1
hmb
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This problem is from Hrbacek and Jech, 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.
 

Answers and Replies

  • #2
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Given any [tex]a\in A[/tex], what properties should the cell of [tex]a[/tex] in [tex]\inf T[/tex] have? How does that tell you how to compute the cell of [tex]a[/tex] in [tex]\inf T[/tex] from the members of [tex]T[/tex]?
 
  • #3
hmb
15
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Is the cell of [tex]a[/tex] in [tex]infT[/tex] the equivalence class of [tex]a[/tex] modulo [tex]E_{infT}[/tex], i.e., [tex][a]_{E_{infT}}[/tex]? If so, then I think the properties the cell of [tex]a[/tex] in [tex]infT[/tex] would have would be as follows:

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

Is that right?
 
Last edited:
  • #4
hmb
15
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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.
 

Attachments

  • #5
hmb
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. . . and here is the last page.
 

Attachments

  • #6
hmb
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Sorry, the 5th-last line on the last page should read:

For every [tex]x \in B[/tex] and [tex]a \in A[/tex], [tex]\left[a\right]_{E_{x}} \subseteq \bigcap \left\{\left[a\right]_{E_{y}} \left| y \in T\right\}[/tex].
 
  • #7
hmb
15
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OK, I think I've got part (d) now as well, although I think the 'hint' was supposed to read: [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].] I have attached the proof.
 

Attachments

  • #8
hmb
15
0
Sorry, another mistake; in the first line it should say [tex]x \leq S[/tex], not [tex]x \in S[/tex].
 

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