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What is the meaning of "Then B = {f(X) ⊂ P(n) : X ∈ A} is nonempty, and has a ⊂-maximal element"?

What is the meaning of "Then B = {f(X) ⊂ P(n) : X ∈ A} is nonempty, and has a-example-1
User RakeshNS
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See my other comment :).

B is the set made of all the sets f(X) where X is a subset of A.

A is a subset of the powerset of F. That means A includes some subsets of F, e.g. X.

Now P(n) is the powerset of the first n natural numbers. So P(n) includes all the possible sets including 1 to n of the first n natural numbers (and the empty set, though not important here).

Now, f is a function that maps any element of f ONE TO ONE to a natural number k.

Then we define f(X), where X={a,b,c,…} (element of A, therefore a subset of F), so f(X)={f(a), f(b),…..} which is a set of natural numbers :)

So subset maximal means, there is some X in A for which f(X) (by definition an element of B) is the biggest element of B by size. They want to differentiate “Maximum size set” from “set with Maximum size element” I think :)
User Amauris
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