Do repeated prime factors count as distinct members in a set for proof purposes?

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SUMMARY

The discussion centers on the treatment of repeated prime factors in set theory, specifically regarding whether they count as distinct members in a set. When defining a set A of prime factors of an integer a, repeated factors do not count as distinct members. Consequently, when forming a set C from the intersection of sets A and B (where B contains the prime factors of another integer b), the repeated prime factors will appear only as many times as they occur in the set with the least frequency. The concept of a multiset is crucial for understanding this topic.

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  • Familiarity with prime factorization
  • Knowledge of multisets and their properties
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Students of mathematics, particularly those studying set theory and number theory, as well as educators seeking to clarify concepts related to prime factors and multisets.

Zeth
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This is for a proof but I was generally more curious so it isn't in the homework section.

If I were to make a set A which is defined as all the prime factors of an integer a there could be some numbers in A which are repeated, would these count as distinct members or not? The reason why I was wondering is if I made another set by the same criteria for an integer b would and then I made another set C with members A cap B would the repeated numbers show up only as many times as they appear in the set that contains them least or most? I think it should be least but I don't have text on the subject yet and we haven't covered much set theory. I would also appreciate some link or recommendation for a text at 1st year university level.
 
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Thanks a lot, that has just the properties I needed for the proof.
 

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