Showing that a set is well-defined

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I think, I would need more information.In summary, the conversation discusses the proof of the existence of a set ##C## that satisfies the condition ##C = \{\{x\} \times B : x \in A\}##. The approach suggested is to use the subset axioms or the axiom schema of restricted comprehension. The argument is presented that if ##C^* = \{w \in \mathcal{P} (A \times B) : w = \{x\} \times B ~\text{for some}~ x \in A \}##, then ##C^*=C##. However, further clarification is needed on the axioms and the meaning of ##\math
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Mr Davis 97
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Homework Statement


Assume ##A## and ##B## are given sets, and show that there exists a set ##C## such that ##C = \{\{x\} \times B : x \in A\}##.

Homework Equations

The Attempt at a Solution


To show that this set exists, I thought to maybe use the subset axioms (or the axiom schema of restricted comprehension). I claim that if ##C^* = \{w \in \mathcal{P} (A \times B) : w = \{x\} \times B ~\text{for some}~ x \in B \}##, then ##C^*=C##. First ##C^*## is well-defined by the subset schema. Additionally, it contains all elements of the desired sort because if we fix ##x \in A##, then clearly ##\{x\} \times B \subseteq A \times B##, so ##\{x\} \times B \in \mathcal{P} (A \times B)##. So ##C^* = C##, and hence ##C## is a well-defined set.

Is this a correct argument? Is there any way I could make it flow better?
 
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It would help a lot, if you
  1. said what exactly has to be shown, since the existence appears to be already given by the fact, that you wrote down the set in a constructive way.
  2. said what the axioms are, e.g. via a link or a reference, especially because I doubt they are everywhere the same ones or that anybody learned them by heart
  3. said, what **** ##\mathcal{P}(A)## is. I suppose it is meant to be the power set of ##A##, but this is not the only way to denote a power set, so a definition would be at least polite
  4. checked, whether there is a typo in your definition of ##C^*##. I assume there is.
At least those where the points which prevented me from having a closer look.
 

1. What does it mean for a set to be well-defined?

A set is considered well-defined when all of its elements can be clearly and unambiguously determined. This means that there is no room for interpretation or confusion when deciding whether an element belongs in the set or not.

2. How do you show that a set is well-defined?

To show that a set is well-defined, you must provide a clear and precise definition of the set and its elements, and then demonstrate that all elements meet this definition. This can be done through logical reasoning, examples, or by showing that each element satisfies certain criteria.

3. Can a set be well-defined if it contains elements that are not explicitly mentioned?

Yes, a set can still be well-defined even if it contains elements that are not explicitly mentioned. As long as the definition of the set is clear and unambiguous, any element that meets this definition can be considered a member of the set, even if it was not initially mentioned or thought of.

4. What happens if a set is not well-defined?

If a set is not well-defined, it means that there is a lack of clarity or precision in its definition, making it difficult to determine which elements should be included in the set. This can lead to confusion and inconsistencies in mathematical reasoning and can potentially invalidate any conclusions drawn from the set.

5. Why is it important to show that a set is well-defined?

Showing that a set is well-defined is important because it ensures the validity and reliability of any mathematical statements or proofs that use this set. It also helps to avoid any potential contradictions or errors in reasoning that may arise from an unclear or ambiguous set definition.

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