A majority of the students are expected to vote in the class election.
This sentence uses the indefinite article before the "majority of the students". This has the effect of introducing this group of students to the reader or the listener.
It may give a hint that the exact number of the students that will vote is uncertain: it could be 51%, but then it could be 88%.
I would also half-expect the following sentences to contain further information on this particular group of students, like this:
A majority of the students are expected to vote in the class election. This is a rag-tag group of A-grade students and 'politically active' D-graders.
Now, to the second sentence.
The majority of the students are expected to vote in the class election.
This sentence seemed not very natural to me, but it is probably okay. It uses the definite article before "majority of the students", as if referring to a known group. But in reality it likely refers to the concept of majority (superiority in numbers).
I've tried to look for similar structures on the web, say "The majority of the citizens", "The majority of the voters", and the results are scarce (about 400 google-hits each). But so are the results for the same phrases with a, so I'm unsure.
Here's one quote apparently coined by a native English speaker:
If I recollect rightly, Aristotle observes, that a democracy has many striking points of resemblance with a tyranny. Of this I am certain, that in a democracy, the majority of the citizens is capable of exercising the most cruel oppressions upon the minority, whenever strong divisions prevail in that kind of polity, as they often must; and that oppression of the minority will extend to far greater numbers, and will be carried on with much greater fury, than can almost ever be apprehended from the dominion of a single sceptre. (Edmund Burke)
Here, the majority is contrasted with the minority, and this probably makes the nouns "definite": we are all accustomed to groups of people being divided into a majority and a minority on numerous matters. This use of the probably stresses that the author talks about definite concepts, not about particular groups of people.
1. The grammar: "partitive structures"
As I understand it, constructions of the type (A/The) Z of X have a partitive effect: we tend to assume that Z is part of X. Hence, the use of the is justified only when the reader is familiar with this particular group Z that is part of X.
The writer can oftener safely presume that all his readers are familiar with abstract concepts than with some particular groups of people or items. Hence, this partitive use of the seems to prevail in phrases speaking about well-known abstract topics:
She is studying the history of Russia. (okay)(see Quirk et al., 5.58)
She has already bought the majority of the Russian history books she needs. (not okay? or marginally okay?)
- The grammar: "collective nouns"
The word majority is a collective noun and could be treated either as a plural noun or as a singular noun. But following a it is usually plural, especially in of-partitive structures:
A majority of the students were polled. (according to the grammar book "Woe is I")
Here's one explanation I've found on the web (Grammar Logs, 2000):
In the expression "a majority of ____," we invariably use a plural verb. The word majority is always singular in itself when it means a superiority in number — "The majority was small but enough to carry the day." In other senses, the word can be either singular or plural, depending on whether you're thinking of the group in question as a singular entity or as a group of individuals: "the majority [of the students] have voted already"; "the majority is sometimes wrong." But "a majority of ______" requires a plural verb.
Here's the Google Ngram comparison for "majority of __ is" vs. "majority of __ are" (kudos to Damkerng T.)
(Bear with me, I'm not a native English speaker and might be mistaken in my apprehension of articles.)
good of you to...
Your actions in a particular situation were appropriate, and maybe exceeded what could reasonably be expected of you.
It was good of you to take Geoff home last night: he was so drunk that I was worried about him!
good for you to...
Your actions are/will be beneficial to you.
It's good for you to get some healthy exercise every day
It is often used in a conditional form:
It would be good for you to get some healthy exercise every day
good to you to...
As this NGRAM shows, this expression is uncommon and, when used in this context, seems to mean the same as the for version.
It occurs more often in sentences like this:
Does it sound good to you to go to the beach tomorow?
Best Answer
"Be careful of keeping her shoes clean" literally means "There are dangers of keeping her shoes clean and you should be cautious about them." But I have never heard it said by people who intended it that way. Every time I have heard that statement made, the person had intended to mean "Make sure she doesn't get her shoes dirty."
As user514428 said, normally careful to is a reminder to make sure something is done, and careful of is looking out for a hazard. But you should also be aware that be careful of is often used in the incorrect manner in which your second example uses it. Humans are not always the most logical thinkers and thus not always the most logical speakers. This is commonly evident in English usage. My studies of other languages never really got anywhere, but I'm pretty sure it's also true for German, Spanish, Latin, Italian, French, and Japanese. I would guess it's true for all of the languages we've made to communicate with each other.
I've been told that it's just a matter of considering what would be reasonable for the sentence to mean. For example, in this case, it would be reasonable to assume that the girl's caretakers would not want the shoes to be dirty, so it's probable that they're trying to warn against them getting dirty. However, I've met enough people to know that while that works in this case, different people are sufficiently different that it doesn't work in all cases.