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Suppose $\alpha$ and $\beta$ are sets. Suppose that the formula $\alpha\supseteq\beta$ is almost obvious.

Now which of the following (true) alternatives make a statement of a lemma (used by one theorem below)?

  1. $\alpha\subseteq\beta$
  2. $\alpha=\beta$
  3. $\forall K\in\alpha:K\in\beta$

In the proof of the theorem we need to prove (among other statements inside the proof) $\alpha=\beta$.

porton
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2 Answers2

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We have defined $\beta \subset \alpha$ so the lemma (a proven statement used as a stepping-stone toward the proof of another statement) is 1) as it helps you in your route to your proof. The third doesn't make logical sense.

Don Larynx
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  • What about the second? – porton Sep 21 '13 at 13:21
  • What is wrong with the third? It seems unnecessary to spell it out like that, but logically it is fine. – Tobias Kildetoft Sep 21 '13 at 13:23
  • @Jossie, the third one's exactly the same as the first one. – DonAntonio Sep 21 '13 at 13:25
  • "For all $K$ in a such that $K$ in b" doesn't make sense, it's missing something at the end – Don Larynx Sep 21 '13 at 13:26
  • The second seems to be nonsensical within the apparently wanted deduction: if we want to reach $,\alpha=\beta;$ , then assuming or proving $,\alpha=\beta;$ already does the job! – DonAntonio Sep 21 '13 at 13:26
  • That is not what seems to be written there, @Jossie, but rather "for all K that is in $;\alpha;$ : K is in $,\beta,$ – DonAntonio Sep 21 '13 at 13:27
  • In that case it makes more sense, I just take the colon symbol to always mean "such that" – Don Larynx Sep 21 '13 at 13:28
  • I have never taken the colon to mean that, but I'm not a mathematical logic guy, so... – DonAntonio Sep 21 '13 at 13:28
  • Perhaps in case one needs a strict logical formula, isn't it worth writing it like $\forall K [K\in\alpha \to K\in \beta]$ (or, i don't know, $\forall K\in \alpha [K\in \beta]$)? Concerning this colon symbol, i'd agree with Jossie, since my professor also used it to denote "such that"... – W_D Sep 21 '13 at 13:53
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If what you need later is equality, that is how I would state the lemma, to make references to the fact shorter. If the lemma only mentions one inclusion, you would have to remark that the other inclusion is obvious each time you need it, rather than just once in the proof of the lemma.

(Here I am assuming that you need the equality $\alpha = \beta$ as part of the proof of some later theorem, rather than $\alpha = \beta$ being that later theorem).

Tobias Kildetoft
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