Mathematics · Set Theory

An infinite binary word a is a string (a 1 a 2 a 3 ...), where each a n ∈ {0, 1}. Fix a word s = (s 1 s 2 s 3 ...), where s n = 1 if and only if n is prime. Let S = {a = (a 1 a 2 a 3 ...) | ∃m ∈ &naturals; such that a n = s n , ∀n ≥ m}. What is the cardinality of S?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

Correct answer

Countably infinite

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An infinite binary word a is a string (a 1 a 2 a 3 ...), where each a n &isin; {0, 1}. Fix a word s = (s 1 s 2 s 3 ...), where s n = 1 if and only if n is prime. Let S = {a = (a 1 a 2 a 3 ...) | &exist;m &isin; &naturals; such that a n = s n , &forall;n &ge; m}. What is the cardinality of S?