In the following sentence, the teacher said that, since there is no 从 at the beginning it means that not everyday 我 have exams.
If there was a 从 it would mean that 我 would have exams everyday until 六 月初.
Is that true?
Another question, are 至 and 到 interchangeable in this sentence?. Do they have the same meaning?
我 明年五月底 至 六月初 要 会考。
In my opinion, the sentence means the same thing with/without 从.
Look at this sentence for comparison: 我下周从周一至周三要会考。 Vs. 我下周周一至周三要会考。Both means that you will take exams from Monday to Wednesday. There isn't any difference.
The reason why your sentence (我 明年五月底 至 六月初 要 会考。) can lead different interpretations with/without 从 is that 五月底 and 六月初 are vague, which gives people a rough idea on when your exams would happen (during 五月底 and 六月初). If you put 我 明年五月三十号 至 六月一号 要 会考 or 我 明年从五月三十号 至 六月一号 要 会考, there is no confusion at all. Both mean the same thing.
So, the issue here is not about omitting 从 at all, but 五月底 and 六月初.
[从 X 至 Y] = [from X to Y] (describes a continuous time)
[X 至 Y] = [ between X and Y] (describes a range of time)
Example:
"He visited me from Mondays to Saturday" means he visited me every day from Mondays to Saturday
"He visited me between Mondays and Saturday" means he visited me some days between Mondays and Saturday
五月底 (至) 六月初
- (between) the end of May (and) the beginning of June
- the end of May (to) the beginning of June -- [presume (从) is implied]
~
(从)五月底 (至) 六月初 = (from) the end of May (to) the beginning of June
至 and 到 are interchangeable in this sentence
Since 五月底 至 六月初 can be interpreted as (从)五月底 (至) 六月初 with 从 omitted, it would be more precise to say "在五月底 和 六月初 之間" if you meant to say "between the end of May and the beginning of June"
One more example:
网上订的那东西，(在)五月至六月(之間)到达 (O) -[presume 在 and 之間 are omitted]
网上订的那东西， (从)五月至六月到达 (X) -[cannot presume 从 is omitted, an item cannot arrive every day]
