Feedback of Test1

I was confident on my test even after comparing the answers from Prof. Heap but the marks were really lower than I expected and I couldn't figure out why.
Here is some comparison.
QUESTION 1.
(sample solution)

(my solution)
 












 
Since question 5 and 8 are so similar, I put the explanation in the top after '#' to reduce redundant words. (The idea comes from programming). Apart from that, I think my solution is identical with the sample solution. For question 7, my idea was if 'for all' (q2) is satisfied,  then 'there exists'(q3) must be satisfied as well.
I couldn't see the flaw in my logic and that's what confused me the most.

QUESTION 2.
(sample solution)






 
(my solution)






Indeed, I should have stated the connection between Delta and Epsilon precisely as sample solution did. But if I choose Delta = Epsilon(for both of them greater than 1), the statement is also satisfied.
Sometimes, I find explanation is the hardest part in the exam or assignment. I have tried my best to express myself as precise and clear as I could but still lose marks for that.

The Order of Multiple Quantifiers

Thoughts about switching the order of quantifiers.
For example,  for the statement below:

Can we switch the first two quantifiers?
At first I kept thinking about Calculus and it was really distracting. Then I realized I can simply ignore the implication part and focus on the quantifier itself.
Then I thought whether 'there exists a Delta in natural numbers' is in the front or the second, we are going to pick delta first to evaluate the statement. Therefore, there should be no difference if I switch the first two quantifiers.
However, having discussed with TA, I realized there is a difference and the way he showed me was insightful. He further simplified the questions by giving Delta and Epsilon actual meanings:
Let C be courses, S be students, L(x, y) be x likes y.
Statement1: there exists s in S, for all c in C, L(s, c)
Statement2: for all c in C, there exists s in S, L(s, c)
Are S1, S2 the same? No.
S1 is saying that there is a student likes all courses but S2 means that for all courses there is at least one student likes it (that student need not like all courses).
Problem solved.