Prove that the set of all infinite bitstrings is uncountable.Sol.pdf
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The document proves that the set of all infinite bitstrings is uncountable using diagonalization. It outlines the steps of assuming the set can be enumerated, creating an arbitrary list of its elements, and constructing a new element that contradicts the list. This contradiction demonstrates that the initial assumption was false, confirming the set is uncountably infinite.
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![Prove that the set of all infinite bitstrings is uncountable.
Solution
We used diagonalization to show that the set of real numbers in the interval [0, 1] is
uncountable, i.e. uncountably infinite. The technique was a follows: (1) Assume that a set S can
be enumerated. (2) Consider an arbitrary list of all the elements of S. (3) Use the diagonal from
the list to construct a new element t. (4) Show that t is in S but is different from all elements in
the list and so is not in the list. Contradiction. This shows that the original assumption that S is
countable was false, so S is uncountably infinite.](https://image.slidesharecdn.com/provethatthesetofallinfinitebitstringsisuncountable-230326103700-da67c002/85/Prove-that-the-set-of-all-infinite-bitstrings-is-uncountable-Sol-pdf-1-320.jpg)
Prove that the set of all infinite bitstrings is uncountable.Sol.pdf
- 1. Prove that the set of all infinite bitstrings is uncountable. Solution We used diagonalization to show that the set of real numbers in the interval [0, 1] is uncountable, i.e. uncountably infinite. The technique was a follows: (1) Assume that a set S can be enumerated. (2) Consider an arbitrary list of all the elements of S. (3) Use the diagonal from the list to construct a new element t. (4) Show that t is in S but is different from all elements in the list and so is not in the list. Contradiction. This shows that the original assumption that S is countable was false, so S is uncountably infinite.