Assignment 1 some tricky questions

So we are mainly working on Assignment 1 this week.

Overall this assignment is pretty easy, but there are some tricky questions.

Question 1

(a) All acronyms are catheterized unless they are bifurcated.
At first I thought the sentence means there are two types of acronyms: catheterized and bifurcated. It shows that these two have no intersection part which means there is no such acronyms that is catheterized and bifurcated. So when we negate it, it will be there exists an acronym which is catheterized and bifurcated.  Then in symbolic form is ∃a∈A,B(a)⋀C(a)

When I look at this question again, I find this is the right way which means ALL acronyms are catheterized and bifurcated.  So when we negate it, it will be there exists an acronym which is non-catheterized and non-bifurcated.

(e) For an acronym to be catherized, it is necessary and sufficient that it be diagonal.
According to the lecture slides of week 2, we now that " For P to be true, Q is necessary."
In this question P: an acronym to be catherized Q: it be diagonal
For an = For every = All the negation will be there exist an acronym which is catherized and non-diagnonal or non-catherized and diagonal. Then in symbolic form is
∃a∈A,(D(a)∧¬C(a)) ⋁ (C(a)∧¬D(a))

Question 3

(a) One, and only one, student in X is more popular than Zorn.
When I look at the question the first time, I got confused by "One and only one", then I thought about it, "one and only one" simply means "there exists one student x who is more popular than Zorn".

∃x∈X,(P(x,Zorn)∧S(x))∧(∀y∈X,(S(y)∧P(y,Zorn))⇒EQ(x,y))