# Translating a sentence into a predicate formula

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.
• Do you know quantifiers ? Like e.g. $\exists x$ – Mauro ALLEGRANZA Jan 23 '19 at 16:18
• How are you asked to mange the "n" ? Do you know "numerical" quantifiers ? if not, Try with the simple cases : $n=1$ and $n=2$. – Mauro ALLEGRANZA Jan 23 '19 at 16:30
• How we have to read "besides possibly himself" ? That we have to exclude himself from counting, I think... – Mauro ALLEGRANZA Jan 23 '19 at 16:31

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