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Discrete math -- 2

Define ?: ?(ℕ) ∖ {∅} → ℕ by setting ?(?) to be the smallest element of ? whenever ? ∈ ?(ℕ) ∖

{∅}. (Note that ?(?) denotes the power set of ?.)

Prove or disprove the following:

(a) The function ? has an inverse.

(b) ?−1({?}) is uncountable for some ? ∈ ℕ.

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