Juanito, we try not to do homework assignments for you, but if you'd like I'll show you how "either" and "neither" are used.
"either" means, “one or the other of two”.
"neither" means, "not one or the other of two".
"either" and "neither" are are called correlative conjunctions; they work in sets:
either . . . or . . .
neither . . . nor . . .
They are also singular in number, so they take a singular verb. For example,
Either Sam or Max has the best grade in the class. (singular verb)
Neither Sam nor Max has the best grade in the class. (singular verb)
But there's an acceptable exception. If there's an "of" phrase with a plural noun (e.g., 'of the candidates'), some speakers might choose a plural verb. The reason being, 'candidates', a plural noun, is closer to the verb than is singular 'Neither'.
Neither of the two candidatesare running next year. (plural verb)
Neither of the two candidatesis running next year. (singular verb)
It'd be best to reword the sentence, like this,
Neither candidate is running next year.
None of the three candidates is running this year.
"none" refers to three or more, whereas "neither" and "either" refer only to two.
If the subjects do not agree in number, like,
Neither you nor I _____ going.
the rule is to use a verb that agrees in number with the closest or nearest subject, like this,
Neither you nor I am going.
Neither I or you are going.
none has baggage. It hails from Old English (OE): adjective ne (no) + noun an (one). There were two forms: one expressed a singular sense, the other inflected to express a plural sense, but with loss of inflection, the two forms became homophonous, leaving,
Singular none: not one of (several)
Plural none: not any of (several)
Traditionalists, those who adhere to the rules of the grammar, argue none (which by the way is a pronoun) expresses "less than one", so its verb should be singular in number. Logical? Well, I'm not a traditionalist, but their argument stands: none, as you know, is a combination of two words, "no", an adjective, and "one", a noun. Since the noun is singular its verb should be singular, too. That's the Modern English rule, right? One problem with the argument, though, is this: none's dual usage didn't "die out" with its loss of inflection: speakers and writers continue to use both, singular and plural; they taught and teach their children, albeit indirectly, and so on, and so on, as language lives.
Descriptivists, on the other hand, those who describe the rules that speaker actually use, say that none is used with singular and plural verbs, especially when none is followed by an "of" phrase; e.g., "None of them have/has been to that movie" - that isn't a new trend, either. The oldest known documented usage goes back to the Middle Ages (See here for weighty sources).
As for Current English, my Oxford (same title) says, "the verb following none of . . . can be singular or plural according to the meaning. They give "not any one of . . . ", but I feel they've erred slightly in their example. I'll show you what I mean. The examples that follow are mine:
Plural: None of them are (not any of several)
Singular: None of them is (not one of several)
For the singular sense, 'one' is implied, whereas for the plural sense, 'any' is implied. "any" doesn't express 'one' - Rather, it agrees in number with whatever it's modifying; e.g., any book, any books.
In short, given the definition "any of several", 'any' modifies 'several', so the phrase expresses a plural sense. If "any of several" means, "none", then "none" can in fact express a plural meaning.
Singular none: not one of several OR not any one of several
=> 'any' modifies 'one', making the phrase singular in number.
EX: None of them has seen the movie.
Plural none: not any of several, NEVER not any one of several
=> 'any' modifies 'several', making the phrase plural in number.
EX: None of them have seen the movie.