Exercise · ZW-CANTOR-S-DIAGONAL-THEOREM-007 · ★★☆ Intermediate
What does the diagonal argument imply for \(\mathbb{R}\)?
Choose one option.
Show answer
- B. \(\mathbb{R}\) is not countable.
Explanation
Binary expansions let the argument carry over to \([0,1]\), and from there to \(\mathbb{R}\). The other statements are false.
Review the concept: Cantor's diagonal theorem
Theorem proving that the set \(\{0,1\}^{\mathbb{N}}\) of binary sequences is not countable, by building a sequence that appears in no list.
\[t(n)=1-s_n(n)\]
More practice problems: cantor's diagonal theorem
- ZW-CANTOR-S-DIAGONAL-THEOREM-001 What does Cantor's diagonal theorem state?
- ZW-CANTOR-S-DIAGONAL-THEOREM-002 Which sequence does the diagonal proof build to escape a list?
- ZW-CANTOR-S-DIAGONAL-THEOREM-003 Is ℕ countable?
- ZW-CANTOR-S-DIAGONAL-THEOREM-004 Which of these sets is uncountable?
- ZW-CANTOR-S-DIAGONAL-THEOREM-005 With s_1=000…, s_2=111… and s_3=0101…, what are the first three terms of the diagonal sequence t, w…
- ZW-CANTOR-S-DIAGONAL-THEOREM-006 Why can the diagonal sequence t not appear in a list s_1,s_2,dots?
- ZW-CANTOR-S-DIAGONAL-THEOREM-008 Which of these statements is true?
- ZW-CANTOR-S-DIAGONAL-THEOREM-009 Let A be the set of binary sequences with finitely many ones. Is A countable?
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