What's the difference between a few, few, the few? Which one is formal or informal?
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The few is quite rare, and used only when designating a small number of things or people that have some special distinction
- the few rounds left in their magazines, the few guests who escaped the fire
There's a big difference between few and a few, however.
They're directed quantifiers, and they point in opposite directions.
- a few means 'a small, but still positive, number', while
- few is a negative quantifier, and means 'fewer than expected, predicted, or wished'
Few governs negative polarity items like ever, producing the following pair:
- Few people ever come here in the winter, but not ...
- *A few people ever come here in the winter.
because a few isn't negative and therefore can't trigger ever;
as well as the following pair, which produce opposite results:
- Thank heavens that few people were hurt!
- Thank heavens that a few people were hurt!
because you're thanking heavens for negative versus positive injuries.
Negatives are pretty tricky; much trickier than most people expect.
In terms of your first two examples, which are really related to this concept of how you should use 'a few' or simply 'few': -
If you say
"I have a few problems"
it is different in meaning to saying
"I have few problems"
In the first you could read it as "I have some problems"
In the second the emphasis is on the 'few' - so you could read it as "I only have a few problems". Which is a feature worth feeling good about, as opposed simply having "a few problems" which really isn't such an esteemed position to be in.
When you omit the 'a' you are emphasizing the 'few' in the sentence.
Regarding your last question, and when you should use 'the' few. This a point of grammar. You might want to search for indefinite or definite articles on the boards here. Or start by looking at this related question: