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Problem 3.40. (a) Translate the following sentence into a predicate formula:

There is a student who has e-mailed at most n other people in the class, besides possibly himself.

The domain of discourse should be the set of students in the class; in addition, the only predicates that you may use are

  • equality,
  • E.x; y/, meaning that “x has sent e-mail to y.”

(b) Explain how you would use your predicate formula (or some variant of it) to express the following two sentences.

  1. There is a student who has emailed at least n other people in the class, besides possibly himself.
  2. There is a student who has emailed exactly n other people in the class, besides possibly himself.
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    $\begingroup$ Do you know quantifiers ? Like e.g. $\exists x$ $\endgroup$ – Mauro ALLEGRANZA Jan 23 at 16:18
  • $\begingroup$ How are you asked to mange the "n" ? Do you know "numerical" quantifiers ? if not, Try with the simple cases : $n=1$ and $n=2$. $\endgroup$ – Mauro ALLEGRANZA Jan 23 at 16:30
  • $\begingroup$ How we have to read "besides possibly himself" ? That we have to exclude himself from counting, I think... $\endgroup$ – Mauro ALLEGRANZA Jan 23 at 16:31
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Solution to part (a): $ \exists s \Big((\forall u ^\neg E(s, u) ) \land \big( \exists x\exists y (x\ne y \land y \ne s \land x \ne s) \land (E(s, x) \lor E(s,y)) \land \forall z ((z \ne x \lor z\ne y \lor z \ne s) \implies ^\neg E(s,z))\big) \Big) $

  • $(\forall u ^\neg E(s, u) )$ stands for the case that s does not send any one an email.
  • $ E(s, x) \lor E(s,y)$ stands for 2 people
  • $\forall z ((z \ne x \lor z\ne y \lor z \ne s) \implies ^\neg E(s,z))$ stands for at most 2 people
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  • $\begingroup$ @MauroALLEGRANZA Sir, may you check whether this solution is correct? $\endgroup$ – 王文军 or Wenjun Wang Mar 14 at 7:59

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