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I am reading Michael Sipser's book Introduction to the Theory of Computation, which mentions the set $$S = \{ n \mid \text{$n$ is an integer and $n = n + 1$}\}.$$ This doesn't make any sense to me. I would understand if $n$ were equal to infinity or something, so it probably wouldn't matter if we added $1$ to it. Am I understanding it correctly? Or is this just an empty set?

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2 Answers 2

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But infinity isn't an integer. Since there is no integer $n$ such that $n=n+1$, you're right that the set is empty.

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This is an empty set which is usually denoted by {}. In this case, the integer n is equal to itself while incremented by 1 which is mathematically impossible. Thus, this is an empty set.

EX: n = 2, then n = n + 1, 2 = 2 + 1. This is impossible.

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    $\begingroup$ Why those downvotes ? $\endgroup$
    – user16034
    Commented Sep 3, 2023 at 18:35
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    $\begingroup$ This is an ancient question that already had a better answer. $\endgroup$
    – Kai
    Commented Sep 4, 2023 at 4:23
  • $\begingroup$ (downvoters please comment what in particular makes a post not useful.) $\endgroup$
    – greybeard
    Commented Sep 4, 2023 at 5:25

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