Set Theory Problem Involving Partitions

In summary, Hrbacek and Jech state that a relation \leq is defined in a partition Pt(A) if and only if for every C \in Pt(A) there is D \in Pt(A) such that C \subseteq D. Furthermore, infT exists if and only if for every a \in A there is an upper bound x for all x in T such that x \leq a. The attempt at a solution for part (c) states that for every x \in B and a \in A, \left[a\right]_{E_{x}} \subseteq \bigcap \left\{\left[a\right]_{E_{y
  • #1
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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.
 
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  • #2
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
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?
 
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  • #4
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.
 

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

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  • #6
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
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.
 

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

1. What is a partition in set theory?

A partition in set theory is a way of dividing a set into non-overlapping subsets, so that every element in the original set belongs to exactly one subset. Essentially, it is a way of categorizing or grouping the elements of a set.

2. How is a partition different from a subset?

A subset is a group of elements that are all part of a larger set. A partition, on the other hand, is a division of a set into subsets. This means that every element in a partition belongs to exactly one subset, while in a subset, an element can belong to more than one subset.

3. What is the importance of partitions in set theory?

Partitions are important in set theory because they allow us to study the structure and relationships within a set. They also help us to understand the properties of a set and can be used to solve complex problems in mathematics and other fields.

4. How do you determine the number of partitions in a set?

The number of partitions in a set is equal to the number of ways in which the elements of the set can be grouped into non-overlapping subsets. This can be calculated using the Bell numbers, which represent the number of ways in which a set can be partitioned.

5. Can partitions be applied to infinite sets?

Yes, partitions can be applied to infinite sets. However, in infinite sets, the number of partitions is generally infinite as well. This means that it is not always possible to determine the exact number of partitions in an infinite set, but we can still use partitions to study the structure and properties of the set.

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