Exercise from Naive Set Theory by Halmos

In summary, Halmos gives an exercise in Section 6 (page 23 in my book) that asks for an intrinsic characterization of sets of subsets of A that correspond to some order in A. The exercise is to find a set C of subsets of A that satisfies certain conditions, such as having the same number of elements as A. However, having the same number of elements is not enough to characterize these sets, as shown by the example of {a}, {b}, {c}, {d}. Therefore, the characterization must also consider the order of elements in the set.
  • #1
tmbrwlf730
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For those who have read Halmos, in Section 6 Ordered Pairs (page 23 in my book), he gives a non-trivial exercise to find an intrinsic characterization of those sets of subsets of A that correspond to some order in A. I'm curious what that characterization is.

A is suppose to be a quadruple {a, b, c, d} and he gives the order c, b, d, a as an example.
C is a set who's elements are sets that for each particular spot in the ordering, that set's elements are those that occur at or before the spot.

So we can write the order in the example above as {c}, {c, b}, {c, b, d}, {c,b,d,a}
and C = { {a, b, c, d}, {b, c}, {b, c, d}, {c}}.

Is the characterization just that C has the same number elements as A?
 
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  • #2
tmbrwlf730 said:
Is the characterization just that C has the same number elements as A?

That is of course necessary but it doesn't characterize these sets. For example

[tex]\{\{a\},\{b\},\{c\},\{d\}\}[/tex]

also has the same number of elements but doesn't correspond to some order in ##A## (I assume you mean a total order).

Look at the set ##C##. Is the set ##C## ordered in some way?
 

1. What is the definition of an exercise in naive set theory?

An exercise in naive set theory is a problem or task designed to help individuals understand and apply the concepts and principles of set theory. It typically involves manipulating and reasoning about sets, elements, and operations on sets.

2. How do exercises in naive set theory help in understanding the subject?

Exercises in naive set theory help in understanding the subject by providing hands-on practice and application of the concepts and principles learned. They also help in identifying and addressing any misunderstandings or gaps in knowledge.

3. Are there different types of exercises in naive set theory?

Yes, there are different types of exercises in naive set theory, including basic exercises, advanced exercises, word problems, and proof-based exercises. These types vary in difficulty level and focus on different aspects of set theory.

4. How can one approach solving exercises in naive set theory?

One can approach solving exercises in naive set theory by first understanding the given problem and identifying the relevant concepts and principles. Then, using logical reasoning and knowledge of set theory, one can apply the appropriate operations and techniques to arrive at a solution.

5. Can exercises in naive set theory be used to improve problem-solving skills in other areas?

Yes, exercises in naive set theory can help improve problem-solving skills in other areas by developing logical thinking, critical analysis, and deductive reasoning abilities. These skills are transferable and can be applied in various academic and real-life situations.

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