I've been doing some practice questions in the textbook for an upcoming predicate calculus lecture and I think I've managed to get A and B (possibly C) correct but I am clueless on how I can manage D and E.
For A I got: ∃x(R(x)∧¬B)
For B I got: ∀x(¬R(x)∧B)
For C I got: ∃x(R(x)∧(x=y)) - Not sure if this is even remotely correct
R(X) : ”X is rich”
X = Y : ”X equals Y”
B : ”Bill Gates”
a) Someone besides Bill Gates is rich.
b) Bill Gates alone is rich.
c) At least two people are rich.
d) Exactly one person is rich.
e) Exactly two persons are rich.
The universe of discourse is the set of all people.
a) $\exists x(R(x) \wedge \neg(x = B))$
b) $\forall x(R(B) \wedge (\neg(x = B) \implies \neg R(x)))$
c) $\exists x (\exists y(R(x) \wedge R(y)))$
d) This one is more or less like b) $\exists x(\forall y(R(x) \wedge (\neg(x = y) \implies \neg R(y))))$
I think I will leave e) to you, since it follows the same idea as b) and d), try it!