difference between “a set of” and “the set of”

This may be obvious to some people, but not to some students who signed up for a math lecture in English, in a country that doesn’t speak English.

The set of positive integers is the set of all positive integers. A set of positive integers is a subset of the set of positive integers.

Example sentences:

“A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.”

The set of all men is a proper subset of the set of all people”

“How can I prove that a set of natural numbers always have a minimum?”

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