Using Rule of Inference to determine if conclusion is valid

Question

Given:

• 1. Every student has an email account
• 2. Maggie does not have an email account
• 3. Homer is a student

Using E(x): x has an email, S(x): x is a student and M to represent Maggie while H represents Homer, I came up with the following premises:

• 1: ∀x[S(x)→E(x)]
• 2: ¬E(M)
• 3: S(H)

I then have to determine if the two following conclusions are valid. The first conclusion:

• Maggie is not a student

I determined this not a valid conclusion because you can only reach it by using the 'denying the antecedent' fallacy. Is this correct?

The second conclusion:

• Homer does not have an email account

I came to the verdict that this is not a valid conclusion by:

• ∀x[S(x)→E(x)] (Premise)
• S(H) (Premise)
• S(H) --> E(H) (UI and 2nd premise)
• E(H) (Modus ponen applied to (2) and (3).

Any help is appreciated.

Thanks

Answer 1

No, Maggie is not a student.
If Maggie were a student, she would have an email account.
But she does not have an email account.
Therefore she is not a student.