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Proposition If $M$ is a finite set of men and $a, b\in M,$ then $a$ and $b$ are equal. (1)

Proof by induction on the number of men in $M:$

  • (1) is true if $M$ contains only one man.
  • Assume that (1) is true for all sets of $n$ men. Let $M$ be a set of $n+1$ men with $a, b\in M.$ $M_a = M \setminus \{a\}$ and $M_b = M \setminus \{b\}.$ These sets have $n$ elements. Let $c\in M_a\cap M_b.$ Then, by the induction hypothesis, $c$ and $a$ are equal, so are $c$ and $b.$ Thus, so are $a$ and $b.$

What's wrong?

Ref. Analysis I, Herbert Amann, p. 45

  • Why are people downvoting? – Lab Oct 06 '21 at 10:27
  • idk. please stop downvote as i'm trying to reach 50 reputation to be able to make comment. –  Oct 06 '21 at 10:28
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    Must have been asked many times here , perhaps with other statements that claim basically the same. The catch is that the truth for $n=1$ does not imply the truth for $n=2$. In fact, if the claim were true for $n=2$, the induction would work and the claim would actually be true. – Peter Oct 06 '21 at 10:28
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    I don't know if that's true, but it should be marked duplicated instead of getting downvotes. Either way I don't think it's really THAT a bad problem. – Lab Oct 06 '21 at 10:33
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    i searched before posting but maybe they are in different forms that i didn't recognize. here i quoted nearly the exact form from the book. –  Oct 06 '21 at 10:35
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    @Lab I agree with you. I am very deceived by many nonsense downvotes. – Jean Marie Oct 06 '21 at 11:41
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    It's usually presented as a proof that all horses are the same color. https://math.stackexchange.com/questions/504432/question-about-horses and https://math.stackexchange.com/questions/428151/questions-on-all-horse-are-the-same-color-proof-by-complete-induction and https://math.stackexchange.com/questions/1957950/all-horses-are-the-same-color-variation and https://math.stackexchange.com/questions/2222480/show-that-all-horses-are-of-the-same-color and https://math.stackexchange.com/questions/1298593/mathematical-induction-horses-made-me-question-my-understanding – Gerry Myerson Oct 06 '21 at 12:05
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    And there are many, many more on this website, including my contribution, https://math.stackexchange.com/questions/43007/what-are-some-examples-of-induction-where-the-base-case-is-difficult-but-the-ind/43139#43139 – Gerry Myerson Oct 06 '21 at 12:12
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    too many duplicates of horse. sorry! when i read this problem, the book mentions about Thomas Jefferson's assertion "all men are created equal", so famous especially to Vietnamese like me that I felt the induction proof awesome. –  Oct 06 '21 at 12:13
  • yes, thank you for your kind hospitality! –  Oct 07 '21 at 23:20

1 Answers1

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Saying "let $x\in X$" is only valid if $X\neq \emptyset$.

5xum
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  • if so, the assumption is wrong, so the statement is still true regardless the conclusion, isn't it? i thought a logic statement is wrong only in the case where the assumption is true and the conclusion is wrong –  Oct 06 '21 at 10:28
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    Note that $M_a \cap M_b$ might be $\emptyset$. – Lab Oct 06 '21 at 10:29
  • when i first read this answer i thought of the proposition itself. now thank to Lab's and Peter's comments, i see that in the case of two elements, there is no $c$ at all. thanks all! when i first read the problem in the book, i thought there is always $c,$ didn't think of this case carefully. –  Oct 06 '21 at 10:59