Translating a sentence into a predicate formula

Question

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.

Answer 1

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