A question about a question on A

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In summary, the theorem states that for a cardinal number n, it is possible to define n! as the cardinality of the set of all permutations of a set of size n. This definition is consistent because the cardinality of any set of permutations will always be equal to n!, making it a logical and valid definition.
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InvisibleBlue
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For those who are not familiar with the notation, A! is the set of all permutations of the set A (permutation being defined as a bijection A~A from the set A to itself).

There is a theorem in my notes which says "For cardinal number n it makes says to define n! = |A!| for some set A such that |A| is n."
I'm trying to prove this theorem (missed my lecture :P) and I don't understand what it means by "it makes sense"? Surely if we're defining n! (where n is a cardinal number, not a natural number) then it can be anything we want and therefore it will make sense because we have defined it to.
 
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It is true that we can define n! to be any number we want, but the point of the theorem is that it makes sense to define n! as the cardinality of the set of all permutations of a set of size n. The reason why this makes sense is because the cardinality of A! will always be equal to n!, no matter what set A is, since the permutations of a set of size n is always n!. So defining n! as |A!| is a consistent definition and thus it makes sense.
 

1. What is "A question about a question on A"?

"A question about a question on A" refers to a specific question or inquiry that is being asked about a previous question or topic that has been discussed on a platform or in a conversation.

2. Why is it important to ask questions about questions?

Asking questions about questions helps to clarify and deepen understanding of a topic or issue. It also allows for critical thinking and promotes curiosity and exploration.

3. How can asking questions about questions improve scientific research?

By asking questions about questions, scientists can identify gaps in knowledge and develop more focused and impactful research questions. It can also lead to new insights and discoveries in the scientific field.

4. Are there any specific strategies for asking effective questions about questions?

Yes, some strategies for asking effective questions about questions include: using open-ended questions, avoiding leading or biased questions, and actively listening and following up with additional questions.

5. Is there a limit to the number of questions that can be asked about a question on A?

No, there is no limit to the number of questions that can be asked about a question on A. However, it is important to consider the relevance and depth of the questions being asked in order to have a productive and meaningful conversation.

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