The set of all sets of natural numbers is called the power set of the natural numbers.
To understand this, let's break it down:
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Natural Numbers: These are the positive whole numbers starting from 1 (or 0, depending on the convention). Commonly represented as N = {1, 2, 3, 4, ...}.
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Set of Natural Numbers: Any collection of natural numbers forms a set. Examples include:
- {1, 2, 3}
- {2, 4, 6, 8, ...} (the set of even natural numbers)
- {1, 3, 5, 7, ...} (the set of odd natural numbers)
- {1}
- {} (the empty set, which contains no elements)
- N = {1, 2, 3, 4, ...} (the set of all natural numbers)
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Power Set: The power set of a set S is the set of all possible subsets of S, including the empty set and the set S itself. It's denoted as P(S) or 2S.
Therefore, the "set of all sets of natural numbers" is the power set of the set of natural numbers, denoted as P(N) or 2N. This set contains every possible combination and arrangement of natural numbers into sets.
Example:
If we consider a small set of natural numbers, say S = {1, 2}, then the power set P(S) would be:
P(S) = { {}, {1}, {2}, {1, 2} }
This shows all the possible subsets that can be formed from the set {1, 2}.
The power set of the natural numbers, P(N), is an uncountably infinite set, meaning it is "larger" than the set of natural numbers itself. This is a consequence of Cantor's theorem, which states that for any set S, the power set of S (P(S)) has a strictly greater cardinality than S itself.
In summary, the answer to the question is the power set of the set of natural numbers, representing the collection of every possible subset that can be formed from the natural numbers.
