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Let A be a set, which of the following is grammatically correct

  1. an element in A, elements in A, a subset in A, subsets in A, or
  2. an element of A, elements of A, a subset of A, subsets of A?

I have seen both used in the literature, but I am not sure if they are used correctly. If both are grammatically correct, is there any difference in the meaning?

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When the context is about mathematical sets, the term elements of A is commonly accepted.

Two sets A and B are said to be disjoint, if no element of A is in B and no element of B is in A.

However, for other types of sets elements in A is also acceptable.

When you are referring to the number of elements, you say: the number of elements in A:

The number of elements in set A is 6.

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