Set-Theoretic Notation Kuratowski

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In summary, the conversation is discussing the Kuratowski definition of ordered pairs and how it can be used to represent <a, b, c> in terms of set-theoretic notation. The value of |<<a, b>, c>>| is then determined to be 3. Lastly, it is shown that <<a, b>, c> ≠ <a, <b, c>> by using the Kuratowski definition.
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bird34
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I'm not sure if I'm on the right track with any of this, but any insight would be greatly appreciated. I have worked out what I believe to be the correct answer, but I really don't have any idea.

Express <a, b, c> =df <<a, b>, c> in terms of the set-theoretic notation only, using the Kuratowski definition.

Given Kuratowski's set-theoretic definition of ordered pairs: <x,y> = df {{x},{x,y}} it seems that the definition of <a, b, c> = df <<a, b>, c> would be <a, b, c> = df{{a}, {a, b}, {c}}

On that construal, what is |<<a, b>, c>>|?

Answer: 3

Show that <<a, b>, c> ≠ <a, <b, c>>.
 
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To show that <<a, b>, c> ≠ <a, <b, c>>, we can use the Kuratowski definition of ordered pairs. For the left side, we have: <a, b>, c> = df {{a}, {a, b}, {c}}. For the right side, we have: <a, <b, c>> = df {{a}, {{b}, {b, c}}}. Since these are two different sets, then it necessarily follows that <<a, b>, c> ≠ <a, <b, c>>.
 

1. What is Set-Theoretic Notation Kuratowski?

Set-Theoretic Notation Kuratowski, also known as Kuratowski's notation, is a system of symbols used to represent mathematical sets and their relationships. It was developed by Polish mathematician Kazimierz Kuratowski in the 1920s.

2. How is Set-Theoretic Notation Kuratowski used?

Set-Theoretic Notation Kuratowski is used in mathematical proofs and set theory to define and describe the relationships between sets. It allows for a concise and standardized way to represent complex set operations and relationships.

3. What symbols are used in Set-Theoretic Notation Kuratowski?

The symbols used in Set-Theoretic Notation Kuratowski include the basic set operations of union (∪), intersection (∩), and complement (¬), as well as the set relations of subset (⊂), superset (⊃), and equality (=). Other symbols, such as parentheses and braces, are also used to denote grouping and set elements.

4. What are some examples of Set-Theoretic Notation Kuratowski?

An example of Set-Theoretic Notation Kuratowski is A ∪ B, which represents the union of sets A and B. Another example is A ⊂ B, which denotes that set A is a subset of set B. Additionally, (A ∪ B) ∩ C represents the intersection of the union of sets A and B with set C.

5. Why is Set-Theoretic Notation Kuratowski important?

Set-Theoretic Notation Kuratowski is important because it provides a standardized and concise way to represent complex set operations and relationships. This allows for clear and efficient communication in mathematical proofs and set theory. It also serves as the foundation for other notation systems used in mathematics and computer science.

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