How Can a Set Both Belong to and Be a Subset of P(N)?

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SUMMARY

The discussion centers on the properties of a set S in relation to the power set P(N) of natural numbers. A correct example for a subset of P(N) is {1, 2, 3}, while a set that belongs to P(N) is {{1}}. Additionally, a set that belongs to P(N) with a cardinality of 5 is {1, 2, 3, 4, 5}. The confusion regarding the definitions of P(N) is clarified, emphasizing that P(N) refers to the power set of all natural numbers, not just the natural numbers themselves.

PREREQUISITES
  • Understanding of set theory concepts, particularly power sets.
  • Familiarity with natural numbers and their properties.
  • Knowledge of set notation and cardinality.
  • Basic comprehension of mathematical logic and definitions.
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  • Study the definition and properties of power sets in set theory.
  • Learn about cardinality and how to determine the size of sets.
  • Explore examples of subsets and their relationships to larger sets.
  • Investigate common misconceptions in set theory, particularly regarding notation.
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Students of mathematics, educators teaching set theory, and anyone interested in deepening their understanding of set relationships and power sets.

NecroWinter
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Hi there, here's the question I am given, i will provide the answer that I think is correct, do you mind checking it and possibly pointing out where I am wrong if I am?

Give an example of a set S such that:
a) S is a subset P(N)
b) S belongs to P(N)
c) S belongs to P(N) and |S|=5

here are my answers:

a) {1,2,3}
b) {{1}}
c) {1,2,3,4,5}
 
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Hey NecroWinter and welcome to the forums.

Just to clarify what is the set P(N)? Is this just all the natural numbers or the power set of the entire set of natural numbers?
 
You have the answers for A and B switched.
 

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