Predicate Logic Practice

Question

There are 5 predicate calculus questions I've been working on, I think I've correctly solved the first four, except the last one I have no idea where I'm going with. I'll present my work and the questions below. Any help would be appreciated. The task is to translate the sentences in the language of predicate calculus using the 3 symbols provided:

R(X) : ”X is rich”

X ' Y : ”X equals Y”

B : ”Bill Gates”

a) Someone besides Bill Gates is rich.

ANSWER: ∃x(R(x)∧¬(x=B))

b) Bill Gates alone is rich.

ANSWER: ∀x(R(B)∧(¬(x=B)=>¬R(x)))

c) At least two people are rich.

ANSWER: ∃x(∃y(R(x)∧R(y)))

d) Exactly one person is rich.

ANSWER: ∃x(∀y(R(x)∧(¬(x=y)=>¬R(y))))

e) Exactly two persons are rich.

ANSWER: No clue!

Answer 1

e) Exactly two persons are rich. No clue? Here's a clue then: That says that some $x$ and some $y$ (where $x$ and $y$ are different) are rich, and that anyone $z$ who is rich is either $x$ or $y$ again.

Answer 2

Hint: In your already given answers check the following things:

• c) Is incorrect because you haven't ruled out the possibility that x=y.

• Try to use the contrapositive: $\neg P\to \neg Q\equiv Q\to P$.

After doing that I believe you will be able to solve your problem.