1. He is a student of our school.
2. He is the student of our school.
3. He is the only student of our school.
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#1 is correct. What about #2? Does #2 mean #3? Does #2 mean there is only one student at our school?
1 is fine, although I'd use "at" or "in" instead of "of."
2 is implying that there is only one student in the school, yes. And again, I'd use "at" ...
3 is stating that he is the only student in the school!