A sentence becomes negative by specific words. Most of those have 'no' in it. For example
No
Nobody
No one
Never
Not
Nowhere
...
And yes a sentence without a subject can be negative. Here's an example:
Stop the car! -- positive
Do not stop the car! -- negative
Do your homework!
Don't do drugs!
No X means ‘not even one’ X: not a single member of the entire group X.
No student in the class is 15 years old yet. ... Of the students in the class, none is 15 years old; all of them are younger.
Other divides the group you are talking about into two parts. You identify one of the parts, and what is left is the other part. For instance:
The red books are mine. The other ones belong to Seema. ... Here I am talking about a group of books. I identify one part, all the red books, as mine; everything that is left, the other ones, are yours.
Roy is the youngest boy in his class. He is 12; all the others are 13. ... The group you are talking about is the boys in the class. One is named, Roy; the others are all the boys who are not Roy.
So whenever you see the word other you have to identify what it is contrasted with.
In your sentence “No student is as intelligent as my son”, you are speaking about a group of students. If your son is a student, he is a member of that group, and the sentence contradicts itself: it is not true that no member of the group is as intelligent, because your son is a member and he is as intelligent as himself.
So you should say “no other student”. This divides the group into two parts: 1) your son, the most intelligent, and 2) all the other students who are not as intelligent as he is.
In this case you may cast student as either singular or plural; but as Martze says, whichever you choose, the verb must agree.
No other students are as intelligent as my son.
No other student is as intelligent as my son.
I prefer the singular, because I think of your son being compared to each other student individually; but the plural won't raise any eyebrows.
Best Answer
"everybody" and "everyone" are equivalent, see these links:
https://www.merriam-webster.com/dictionary/everyone
https://www.merriam-webster.com/dictionary/everybody
The normal way to answer in the negative is:
As you can verify with google, "everybody is not here" is much less common than "not everybody is here".
I can see there might be a few cases where you would say "everybody is not here". It's more emphatic and could be spoken with hostility. For example, if half the class is obviously missing, someone could say: