Please help me convert the statements 1,2 and 3 into First Order Logic formulas given the following predicates.
Note: You can also use new predicates.
f(X) = "X is a fruit",
a(X)= "X is an apricot",
p(X) = "X is an pineapple",
b(X) = "X is bad",
k(X,Y) = "X is more knowledgeable than Y"
w(X) = "X is a woman."
s(X)= X is spoilt
(1) apricots and pineapples are bad fruits and spoilt.
My attempt: a(X) ∧ p(X) -> b(X) ∧ f(X) ∧ s(X)
(2) Every fruit other than apricots and pineapples is bad and spoilt.
My attempt: ∀X f(X) ∧ (¬ (a(X) ∧ p(X))) -> b(X) ∧ s(X)
(3) Some women are more knowledgeable than others except than herself.
My attempt: ∃X w(X) -> k(X)
(1) apricots and pineapples are bad fruits and spoilt.
My attempt: $(a(X) \lor o(X)) \implies (b(X) ∧ f(X) ∧ s(X))$
(2) Every fruit other than apricots and pineapples is bad and spoilt.
My attempt: $∀X (f(X) ∧ (¬ (a(X) \lor o(X)))) \implies (b(X) ∧ s(X))$
(3) Some women are more knowledgeable than others except than herself.
My attempt: $∃X\forall Y (w(X)\land w(Y)\land X\ne Y) \implies k(X,Y)$
EDIT my 2nd attempt for (3):
$(∃X\forall Y) w(X)\land((w(Y)\land X\ne Y) \implies k(X,Y))$